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It sounds like the LCL/MCL is tracking differently on the side with the tight or tighter hip.
The ligament is getting pulled off track as a compensation for the tight hip.
Have you seen an improvement with the improved squat positioning?
What is your ankle rom like?
If you have tight hips you may be missing rom at your ankle too.
It could be a combination of the 2.
Squat Quick Test: Is it your Ankles or Hips?
Case Study: Tight Ankles = Bad Squatting
Super Sumo Groin Mob
What have you done to address your hip?