I can't speak for doing so on a lathe Phil, but I will rant on for them gluttons for punishment who are going to do it by hand.
I've made huge errors in the past lapping with a surface plate,
destroyed two no.60 1/2's, a wafer thin welded Bailey no.5, thankfully I made sure I had sense to buy a thick soled n.8 which I also made convex.
I'm not talking a sheet of paper or two hollow either!
(not to mention other things not so important as a hand plane)
This needs to be understood for the folks using a lapping plate, as there is a large amount of misinformation out there.
The longer a plane is, the more material will be removed off of the toe and heel, as it will start to pivot from the point that's getting larger somewhere in the middle,
and this will lead to it going convex across the width next.
(which means possibly hindering the use of the cap iron if its bad, as it would act like a negative camber)
Having a slightly concave plane only needs to have the edges feathered off, if its only concave across the width, then it is very likely that one would need to stop removing metal from the toe/heel on a plane this long.
That means the abrasive must be shorter than the plane is.
lapping will always favor removal of the edges
For the folks still in doubt, then I suggest getting some 4 or 5mm mild steel plate and doing some experiments with a small square, scribble marker over all of a face and try and remove metal only from the middle, or try and lap flat some intentionally filed convexity , and you will see lapping will just copy the profile and polish it.
You will learn the same thing as with wood preparation that the edges are the reference and those contact points are important for the wood to sit flat and not pivot/see saw about from the middle.
See Charlesworth's video again
Tom