Finite jet detection conjecture for MJ-minimal log discrepancies

Let $X$ be a variety of dimension $d$, let $x\in X$, and let $s_m(x)$ denote the invariant associated with the $m$-th jet scheme at $x$, while $\operatorname{mld_{MJ}}(x,X)$ denotes the Mather–Jacobian minimal log discrepancy. For $0\leq\delta\leq d$, consider a bound depending only on $\delta$ and $d$.

Finite jet detection conjecture. There is a number $N_{\delta,d}$ depending only on $\delta$ and $d$ such that, if

$$s_m(x)\geq\delta\geq 0\quad\text{for all }m\leq N_{\delta,d},$$

then

$$\operatorname{mld_{MJ}}(x,X)\geq\delta.$$

This conjecture would provide affirmative answers to the stated positive-characteristic questions about lower semicontinuity and openness of MJ-canonical and MJ-log-canonical singularities, as well as their stability under small deformations, without assuming resolutions of singularities. It is known for $\delta=d-1$, where one may take $N_{d-1,d}=5$; the general case remains open.