A friend of mine recently had the decorators in, and despite covering his 1930s Broadwood Rosewood Piano,
it got Marked by a Hot Coffee Mug . . . .
The French Polisher called in to fix it used a rather ALARMING Technique . . . . . of
Description courtesy of my Mate . . . . .applying a layer of methylated spirits over the affected area and setting fire to it (repeatedly).
No additional French polishing required, just furniture polish and plenty of elbow grease.
A great result - the whole process took about 20 minutes - and not too expensive either.
I would guess that this should be done progressively, by degrees (extinguishing the fire) -
and repeat 'til mark is gone . .
Obviously - this is one of those "proceed at your own risk" operations . . . .
But it does give you somewhere to go if you are feeling Brave/Confident.
The important thing is that It IS do-able - but get a Pro in if you are not confident . . .