In my industry, when we hear “squeak,” we think “stick-slip-stick-slip-stick-slip” happening very fast. I would try Johnson’s paste wax on that area of the wood, followed by buffing with a soft cloth. Do not get it near the levers nor near any brass parts. Some paste car waxes might also work, but they typically have a polishing agent, and often silicone. You don’t want those on your finish. Unless you are sure you’re getting only wax, I would stick with Johnson’s.
No matter what, it won’t hurt anything to try.