Asymptotic conjugation-invariance conjecture for regular word values

Let $S_n$ be the symmetric group and let $\mathbb F_r$ be the free group on $r$ generators. For a word with constants $w\in S_n\ast\mathbb F_r$, let $\mu_w^{S_n}$ be the distribution of its word values, let $\Sigma(\mu)$ be the conjugation symmetrization of a probability measure on $S_n$, and let $d$ be the Kantorovich--Rubinstein metric induced by normalized Hamming distance. A word with constants is regular if it is not conjugate to an element of $S_n$ and every critical constant of $w^2$ is fixed point free. Asymptotic conjugation-invariance conjecture. If $w_n\in S_n\ast\mathbb F_r$ are regular words with constants of bounded length, then

$$\lim_{n\to\infty}d\bigl(\Sigma(\mu_{w_n}^{S_n}),\mu_{w_n}^{S_n}\bigr)=0.$$

The question is whether symmetrization is necessary for asymptotic conjugation-invariance; the source states this conjecture after proving a bound for the symmetrized distribution.