Thanks for the reply. Unfortunately, I change the feet setting often depending on what desk I'm working at, so I can't remove them or immobilize them completely.
For anyone else with the same question and constraints:
Inspired by the grease mod, I smeared each spring in Vaseline and now the ringing/buzzing noise is gone. A reapplication of Vaseline in the future should remediate new buzzing that emerges with time. I also tried using new, corrosion-free springs that I cut up from longer ones, but they still created noise.