Permutation invariance of Hurwitz orbit sizes in $G_6$

Let $G_6=\langle a,b\mid a^3=b^2=1,\ ababab=bababa\rangle$, and let $\ell$ be a positive integer with $x_1,x_2,\ldots,x_\ell\in\{a,b,a^{-1}\}$. For a permutation $\pi$ of $\{1,2,\ldots,\ell\}$, consider the two factorizations

$$(x_1,x_2,\ldots,x_\ell)\quad\text{and}\quad(x_{\pi(1)},x_{\pi(2)},\ldots,x_{\pi(\ell)}).$$

Permutation-invariance conjecture. These two factorizations have Hurwitz orbits of equal size. The conjecture proposes that, in $G_6$, Hurwitz orbit size is unchanged by arbitrarily permuting a factorization whose entries lie in $\{a,b,a^{-1}\}$. The preceding results establish this kind of invariance for reversals under reversible relations, but the stronger invariance under every permutation is only suggested by testing small cases.