Canon zoom "70-200 2.8L is" focusing infinity problem

I'll start out with saying that autofocus problems can be complicated and I'm not the most experienced with being able to diagnose AF problems. I will try to help you to the best of my abilities though.

Based on the behaviors you listed and the potential AF issues I can think of, I believe you might be on the right track. If you don't already know how the AF encoder works, inside the AF assembly there is a sensor which is used to determine the current position that the focus is set to. This sensor is placed up against a clear plastic disk and uses this disk to detect how much the focus moves (I think there might be like tiny marks on the clear plastic disk or something that is uses to detect how much it moves).

I think the problem you described could potentially be explained by a part of this encoder being dirty. I've seen cases with other models of lenses where grease had gotten onto the encoder or the sensor which caused AF issues. Being that your AF issues happens only in very specific conditions and isn't more erratic, it's probably not from dirt/grease on the sensor itself and likely would just be on part of the encoder.

This video guide is of a repair where dust and sand got into the sensor/encoder. Note that the guide is for the non-IS version of the lens, though having disassembled both model of lenses, I can tell you that they are very similar lenses. The AF assemblies for both version of the lenses do have different part numbers, but I think they are nearly identical and you should be able to follow the video guide with no issues to be able to partially disassemble the AF assembly and clean off the encoder. The disassembly of the lens in general will pretty much also be the same, with the main difference being that the IS version has a larger, 3-section main PCB board. You can also reference this guide to help with the disassembly too, as it is for the IS version of the lens (it won't quite get you all the way to the AF assembly though).

I think I've seen a different video guide where they got more into how to clean the encoder, but I don't remember all the details. I'd probably advise against using isopropyl or any other cleaner as it could potentially damage the encoder or sensor. Wiping over the encoder with a microfiber cloth might be all you need to do to clean it off.