Isogonality growth conjecture for arithmetic covers

Let $k$ be a finitely generated field of characteristic $p>0$, let $X$ be a smooth geometrically connected $k$-variety equipped with a GLP $[?\rho$ and associated groups and covers $\Pi_{\overline k}$ and $X_{C\Pi_{\overline k}(n)}$, and let $\ell\neq p$. For a smooth proper geometrically connected curve, its isogonality $\gamma^{iso}_Y$ is the invariant defined by the least integer $d+1$ such that there is no diagram of nonconstant morphisms $Y_{\overline k}\leftarrow Y'\rightarrow B$ with $B$ an isotrivial smooth proper curve and $\deg(Y'\rightarrow B)\leq d$. Isogonality growth conjecture. Assume that $\rho$ is a GLP, $p>0$ and $\ell\neq p$. Then for every closed but not open subgroup $C\subseteq \Pi_{\overline k}$ one has

$$\lim_{n\to +\infty}\gamma^{iso}_{X_{C\Pi_{\overline k}(n)}}=+\infty.$$

This growth statement is proposed as the missing positive-characteristic ingredient for extending the bounded-degree point finiteness theorem: combined with the known growth of the gonality, it would imply finiteness of points of bounded degree on the relevant covers. The source does not provide evidence of a resolution.