Having smashed Marcel Fassler's 2013 record around the Florida circuit on Saturday, Conway set a marginally quicker best time of 1m41.152s in the afternoon session on Sunday to put the #7 Toyota TS050 Hybrid on top for a second day in a row.
The British driver's time was just 0.052s faster than Fernando Alonso in the sister #8 Toyota.
In the morning session, Kamui Kobayashi again put the #7 car clear of the #8 driven by Alonso, making it a clean sweep of best times for the second-placed Toyota crew in the standings.
Leading the non-hybrid LMP1 charge was new SMP Racing recruit Sergey Sirokin, who posted a 1m42.357s in the #17 BR Engineering BR1-AER in the morning to get within half a second of Kobayashi's best.
Another ex-Formula 1 driver, Brendon Hartley, made it a three-four for the Russian squad with his afternoon best time of 1m42.608s in the #11 BR1.
Gustavo Menezes was the quickest of the Rebellion drivers with a 1m43.296s in the #3 R13-Gibson, less than a tenth quicker than Bruno Senna in the sister #1 car.
Formula 2 racer Nyck de Vries put the Racing Team Nederland squad on top of the four-car LMP2 field with a best effort of 1m47.086s in the #29 Dallara P217.
That was less than a tenth faster than Nicolas Lapierre's best time in the #36 Signatech Alpine Oreca 07.
Porsche usurped Ford at the top of the GTE Pro times in the afternoon session as Richard Lietz posted a 1m57.302s best lap in the #91 911 RSR.
Andy Priaulx's best time in the morning in the #67 Ford GT was just over a tenth slower, while Davide Rigon made it three different marques in the top three in the #71 Ferrari 488 GTE.
GTE Am has headed by Matteo Cairoli in the #88 Dempsey-Proton Porsche with a best time of 1m58.702s, some half a second quicker than Aston Martin driver Pedro Lamy.
There was drama for the class points-leading Porsche 911 RSR of Project 1 Racing, which suffered a major fire during the morning session while Jorg Bergmeister was at the wheel.
With the chassis believed to have been written off, the German squad now faces a race against time to be ready for first practice on Wednesday.