I really should do more reading before I answer... So going back to this, Yes, it does. Coningham did in fact have something to do with the LF.IX, albeit by proxy via the LF.V. Here's what Morgan and Shacklady has to say,
"The impetus given to large scale production [of the LF.IX, my addition] came from Air Marshal Coningham, C-in-C Air Forces in the Western Desert. He originally asked for 'massive' supplies of LF.VB/VCs as 80% of his operations were at low altitude."
"There was little point in building and shipping additional Mk.Vs to the Middle East as future Spitfire production was devoted to the Mk.IX, and the result was a low altitude variant to meet requirements."
Not entirely accurate apparently, see above in Mike Williams' post. The HF designation might have come around at a similar time as the LF designation though, but a year earlier, as per a report written at Boscombe Down dated 3 August 1943 titled "Spitfire HF.IX EN524". However, according to Morgan and Shacklady the first "LF.IX" was a converted Mk.VC MA648, which was fitted with a Merlin 66 on 23 May 1943 by Rolls-Royce, which contrasts with the ORB entry above...
A good idea often has several different origin myths. The one I was raised on is that a test of a captured FW190 vs an early Spitfire IX prompted the desire for improved lower altitude performance. I discount that one, as I don't think the time line fits with the actual Merlin 66 development time line. I also tend to discount the Coningham story for the same reason as I don't see any proper account of when he said it, to whom he said it and how it was conveyed to Rolls Royce. In fact, the following will show he would have had to make the request before he actually used the Spitfire in combat.
The origins of the Merlin 66 and 65 are discussed in "Rolls Royce and the Mustang" by David Birch. These versions of the Merlin were developed at the same time, the 65 for the Mustang and the 66 for the Spitfire. The main difference between them was the propeller reduction ratio (.477 for the 66, .42 for the 65).
Birch quotes a letter of June 24, 1942 from Lovsey to Bulman which discusses the subject of the improved supercharger being developed for the 2 stage Merlin and the plan to lower the supercharger gear ratios concurrently with the introduction of the new supercharger. Since the meeting it references took place on June 10 and since the first FW 190 to be tested by the allies was captured on June 23, the testing story can be discounted. The following quotes are from that letter:
"
With reference to your telephone inquiry this morning , in conjunction with the revised supercharger ratios for the improved performance Merlin 61 blower the following points were discussed with DTD during his visit to Derby on June 10th.
Modifications to the Merlin 61 blower by increasing the diameter of the first stage rotor and changes in the in the rotating guide vane and diffuser have resulted in increasing the full throttle height by approximately 2,000 ft., at the expense of a small reduction of the h. p. at the present all-out boost pressure of 15 lb./sq. in. This blower improvement could, therefore, be used for increasing aircraft performance above the full throttle heights, or by adjustment of the supercharger gear ratios could be used for obtaining a substantial improvement in performance below the full throttle heights, while maintaining the present standard Merlin 61 performance power at full throttle."
This clearly shows that Rolls Royce was thinking about reducing the gear ratios from the beginning. Note that the Spitfire IX with Merlin 61 first went into service around this time
At the time this was going on Rolls Royce was putting Merlins into Mustangs. They proposed using the improved supercharger and the same lower gear ratios for the Mustang.
The following are extracts from a letter to Ellor in Detroit dated, July 1942:
"
Proposed Modifications to the Merlin 61 Supercharger and Bendix Carburettor
Since the Merlin 61 was put into production supercharger development has led to several modifications which have resulted in an improved performance.
It seems possible that these modifications might be standardised from the start in America and full details are therefore attached for vour information. Briefly the modifications consist of the adoption of thin vane diffusers in place of the existing wedge vane type, modified circular arc rotating guide vanes and an increase in the first stage rotor diameter from 11.5 ins. to 12.0 ins."
"As a further development of the improved supercharger a set of revised supercharger gears are now being made up (5.78 :1 and 7.06:1 instead of 6.39 and 8.03)."
"The intention is to improve the low altitude performance as much as possible."
The following is a quote from a letter from Lappin to Freeman dated 16th June 1942:
"I hope you were interested in the Mustang performance figures which I sent to your office last week. We have since made a further exploration into this and we think it is not unlikely that we can again improve the performance by improved blowers and reducing the supercharger ratio."
The following is a quote from a letter from Hives to Freeman dated 28
th June 1942:
"The Spitfire 8 and 9 will make a big improvement and we are also developing another version of the '61', which will give its maximum performance at a lower altitude."
The reality that at high altitude a Spitfire with a Merlin 66 could out perform any aircraft it going to face in combat and by quite a margin.
Coningham didn't receive his first Spitfires until May 7 1942 and they didn't go into action until June 1 1942 (The Spitfire's first victory in the Middle East was June 8, 1942). Since Rolls Royce refers to lower gear ratios in the June 10th meeting, RR must have been working on the problem some time before Mk. Vs saw action in the Middle East, therefore Coningham he must have gotten his demands through to the D.T.D some time before he used a Spitfire in combat.
Incidentally all this occurred before anyone thought of cropping the impeller in a Merlin 45. The Merlin 66 predated the Merlin 45M.
Initially the Americans choose not to follow Rolls Royce's advice on lowering supercharger ratios with the V-1650- 3 which powered early P-51 Bs and Cs but eventually did so with the V-1650-7 which powered later Bs and Cs and all Ds. They did however adopt the improved supercharger which makes the V-1650-3 the equivalent of the Merlin 70, not the 63 as is commonly stated.
The evidence points to crediting Rolls Royce with the idea of improving the performance of the new Merlin at lower altitudes. This was a common theme for both the Spitfire and the Mustang. This should surprise no one as they were the ones who thought of putting 2 stage Merlins into the Spitfire and the Mustang in the first place. They knew more about aircraft performance than most organizations.