Conjecture on density of indices computing the limiting minimal log discrepancy

With the notation of the limiting setup, let $W_i$ be varieties with log terminal singularities, and fix $r_1,\ldots,r_k\in\mathbb{R}_{\ge0}$. For each $i$, let $o_i\in W_i$ and let $\mathfrak{a}_{ij}$ be the corresponding ideals; write $W$, $o$, and $\mathfrak{a}_j$ for the limiting objects, and let $I_l\subseteq I$ be the subsets from the limiting construction. Define

$$J:=\left\{i\in I\mathrel{\big|}\operatorname{mld}_{o_i}\left(W_i,\prod_j\mathfrak{a}_{ij}^{r_j}\right)=\operatorname{mld}_o\left(W,\prod_j\mathfrak{a}_j^{r_j}\right)\right\}.$$

Density conjecture. The intersection $I_l\cap J$ is dense for every $l$.

This extends the preceding special case concerning constancy of minimal log discrepancies and is the analogue for minimal log discrepancies of the corresponding density statement for log canonical thresholds. The source does not provide evidence resolving the conjecture.