And it worked! With basic olive oil, rubbed on with a soft cloth, I was able to basically erase all those dreadful white spots. I still can hardly believe it.
The greater challenge was addressing some more serious water damage left on my Great-Grandma's vanity, when we lived in Panama. It was in a storage unit from November - May, with various other containers stacked on top. When we retrieved our stuff, I was faced with a sad vanity top. Condensation from the plastic totes had dripped down and...well...it was more than surface damage.
But with the piano having such a lovely response to olive oil, I figured there was no harm in trying it here too. Since this was much worse damage...I wasn't expecting it to come as clean as the piano front, but it did improve things quite a bit. Who knew plain olive oil and elbow grease would make such a difference?