1) Is that floor perfectly level where the trainer is sitting?

2) If possible, get on a set of rollers using a fork stand. I would expect there to be a lot of back and forth movement of the rear tire on the rollers. The tire marks attest to this, and you can actually feel the motion under you in the rear. Perhaps you will find, with the help of an observer, that your rear wheel motion is more center-to-side, than side-to-side. At the very least, you'd get a better idea as to how much lateral motion there is in your rear wheel.

I have a Kickr Core, set on an oversized training mat on a hardwood floor. On simulated rides with high gradients, I've found that it walks to the right on me. I don't know if that's due to my right leg weakness or high flywheel speeds on descents. it's kind of annoying when my training table is no longer aligned and the back end of the bike has moved over a foot.

Anyone else had this problem? Any ideas on how to fix it? I'm going to try moving the Core's back legs off the mat so the rubber pads sit directly on the hardwood, see how that goes.