Inversion of adjunction for minimal log discrepancies on quotient singularities

Let $(X,\mathfrak{a})$ be a log pair, let $D$ be a normal Cartier prime divisor, and let $x\in D$ be a closed point. Assume that $D$ is not contained in the cosupport of the $\mathbb{R}$-ideal sheaf $\mathfrak{a}$. The minimal log discrepancies of the pair along $D$ and of the restricted pair on $D$ are defined by

$$\operatorname{mld}_x\bigl(X,D,\mathfrak{a}\mathcal{O}_X\bigr)\quad\text{and}\quad\operatorname{mld}_x\bigl(D,\mathfrak{a}\mathcal{O}_D\bigr).$$

PIA conjecture. One has

$$\operatorname{mld}_x\bigl(X,D,\mathfrak{a}\mathcal{O}_X\bigr)=\operatorname{mld}_x\bigl(D,\mathfrak{a}\mathcal{O}_D\bigr).$$

This is the precise inversion of adjunction assertion investigated for quotient singularities. The paper studies it in connection with the reduction of the lower semi-continuity conjecture; the supplied context does not state whether the conjecture is resolved in full generality.