Only read the abstract, but if as I suspect it is using nested directories as "cells" in the "tape", the proof will require directories to be able to nest arbitrarily deep (which maybe some filesystems already permit; but even if all existing filesystems have some finite limit, this would not be considered an obstacle to the result, since it's certainly possible to construct a filesystem where directory nesting level is limited only by storage size). That's because it needs to be able to simulate a Turing Machine, which could read and write an infinite amount of storage.

Then, there just needs to be a way to force find to stop in some finite amount of time -- that's the halting state. I don't know what mechanism they use for that, but if I were trying to do this, I would lean towards looking for a way to make it error out.

I don’t think most modern file systems have any limit to the depth of nested directories, that’s not how directory trees work. There are other limits like the number of objects in the file system. The ability to reference an arbitrary path is is defined by PATH_MAX, which is the maximum string length. You can still access paths longer than string length, just not in a single string representation.