Second auto-correlation separation conjecture for binary words

Let $w,w'$ be two words, and let $\tilde{c}(w)$ denote the second auto-correlation constant defined by the asymptotic second moment of the number of leaves associated with $w$. The paper establishes that differing values of $\tilde{c}(w)$ likewise imply positive limiting-inferior total variation distance. Second auto-correlation separation conjecture. If $w$ and $w'$ are non-isomorphic words, then

$$\tilde{c}(w)\neq\tilde{c}(w').$$

This is the analogous conjecture for the second-moment constants and is presented as open.