The converse from Kollár-hyperbolicity to V-hyperbolicity

Let $X$ be a normal projective variety and let $H\subset \hat\pi_1(X)$ be a closed normal subgroup. Say that $(X,H)$ is Kollár-hyperbolic with respect to $H$ when the corresponding Kollár-hyperbolicity condition holds, and say that $(X,H)$ is V-hyperbolic with respect to $H$ when $H$ has infinite index and, for every $H$-tower $\pi_n:X_n\to X$, every perverse sheaf $P$ underlying a mixed Hodge module, and every $i\ne0$,

$$\lim_{n\to\infty}\frac{h^i(X_n,\pi_n^*P)}{\deg\pi_n}=0.$$

Converse conjecture. If $X$ is Kollár-hyperbolic with respect to $H$, then $X$ is V-hyperbolic with respect to $H$.

The preceding lemma proves the converse implication from V-hyperbolicity to Kollár-hyperbolicity, so this conjecture asserts an equivalence. The paper says that supporting evidence is given in subsequent sections; no resolution is stated.