The reduction-to-characteristic-p multiplier ideal conjecture

Let $X_0=\operatorname{Spec}R_0$ be a variety in characteristic zero, let $\mJ(X_0)$ be the de Fernex–Hacon multiplier ideal, and let $R_p$ and $\mJ(X_0)_p$ denote reductions modulo $p>0$ as in the source. Let $\tau(R_p)$ be the test ideal of the reduction.

Reduction-to-characteristic-$p$ multiplier ideal conjecture. For all sufficiently large primes $p$,

$$\tau(R_p)=\mJ(X_0)_p.$$

This is motivated by the characteristic-zero/positive-characteristic correspondence between multiplier ideals and test ideals, especially for non-$\mathbb{Q}$-Gorenstein varieties. The source presents the asserted equality as natural but gives no resolution.