Algorithmic radical-boundedness conjecture for word maps

Let $w$ be a reduced word. For a finite group $G$, let $\operatorname{Rad}(G)$ be its solvable radical, and let $\mathfrak{p}_w(G)$ denote the probability that the word map associated with $w$ takes a prescribed value as defined in the source.

Algorithmic radical-boundedness conjecture. There exists an algorithm which, on input $w$, decides whether there is a function

$$g_w:(0,1]\rightarrow[1,\infty)$$

such that, for every finite group $G$ and every $\rho\in(0,1]$, the implication

$$\mathfrak{p}_w(G)\geq\rho\quad\Longrightarrow\quad [G:\operatorname{Rad}(G)]\leq g_w(\rho)$$

holds. If such a function exists, the algorithm also outputs a definition of one possible choice of $g_w$.

The claim would give an algorithmic way to determine whether a lower bound on word-map probability controls the nonsolvable part of every finite group. The source explains that this statement follows if the effective Larsen–Shalev conjecture holds.