Finiteness for overlap-free binary words

Prove or disprove that the refined Diophantine exponent of every overlap-free infinite word over a binary alphabet is finite.

Background

The paper notes that the ordinary Diophantine exponent of every overlap-free infinite word is finite, because an overlap-free word cannot contain sufficiently long periodic factors. It expects the analogous finiteness statement for the refined Diophantine exponent, but describes the question as difficult in this generality. The Thue–Morse word provides a principal example of an overlap-free word for which the paper proves a finite upper bound on the refined exponent.