I was about to go with the wood wedge, but decided to look more deeply into the source of the squeak.
I found the unit only squeaked when it was fully up in locked position, which told me that the problem wasn't in the mechanism's designed movement points, but something in that uppermost locked position.
So, I took a bit of silicone spray that I had put onto a paper towel and rubbed the area's surfaces. Voila! Squeak gone!