The adjoint-test ideal correspondence conjecture along a subvariety

Let $(R,\mathfrak{m})$ be a regular local ring essentially of finite type over a perfect field of prime characteristic $p$, and let $I \subseteq R$ be a nonzero unmixed ideal. Let $\underline{\mathfrak{a}}^{\underline{t}}=\prod_{i=1}^m \mathfrak{a}_i^{t_i}$ be a formal combination, where the $\mathfrak{a}_i$ are ideals of $R$ such that $\mathfrak{a}_i \cap R^{\circ, I} \ne \emptyset$ and the $t_i$ are positive real numbers. Set $A:=\operatorname{Spec} R$, $X:=V(I)$ and $Y:=\sum_{i=1}^m t_i V(\mathfrak{a}_i)$. Assume that $(R,I,\underline{\mathfrak{a}})$ is reduced from characteristic zero to characteristic $p \gg 0$, together with a log resolution $\pi:\widetilde{A} \to A$ of $(A,X+Y)$ used to define the adjoint ideal sheaf $\operatorname{adj}_X(A,Y)$. Adjoint-test ideal correspondence conjecture. Then

$$\operatorname{adj}_X(A,Y)=\widetilde{\tau}_{I}(R,\underline{\mathfrak{a}}^{\underline{t}}).$$

This conjecture predicts that the generalized test ideal along $I$ agrees with the adjoint ideal sheaf after reduction from characteristic zero. The paper establishes partial results toward this correspondence, while the full assertion is not established in the supplied text.