I’ve posted this question on Math Stack Exchange, but I want to bring it here too, because 1] the proof seems missing in the literature, although they are some sporadic mentions and 2] maybe it requires more sophisticated topological tools to prove it or disprove it, like bordism arguments that have been used before for some cases of the square peg problem, and these are out of my reach.

The question is:

Is there a square that all of its corner points lie in the spiral

$$r = ktheta quad 0 leq theta leq infty $$

?