Effective Larsen–Shalev conjecture for automorphic word maps

Let $w$ be a nonempty reduced word in $d$ distinct variables, and let $N(w),\eta(w)>0$ be the constants in the Larsen–Shalev conjecture for automorphic word maps.

Effective Larsen–Shalev conjecture. The constants $N(w)$ and $\eta(w)$ can be computed algorithmically from the word $w$ as input.

This is explicitly described as a slightly stronger version of the preceding conjecture. It would provide effective quantitative control for automorphic word maps, and the source uses it as the hypothesis for deriving an algorithmic statement about solvable radicals.