Hodge ideals and m-adic approximation conjecture

Let $X$ be a smooth complex variety, let $D$ be a reduced effective divisor on $X$, and let $k$ be a non-negative integer. For a point $x\in X$ defined by the ideal $\mathfrak m_x$, and for every integer $r\geq 1$, consider reduced effective divisors $E$ satisfying

$$\mathscr{O}_X(-E)\subseteq\mathscr{O}_X(-D)+\mathfrak m_x^{q(r)}.$$

Hodge ideals and m-adic approximation conjecture. There exists a positive integer $q(r)$ such that, for every such $E$,

$$I_k(E)\subseteq I_k(D)+\mathfrak m_x^r.$$

Here $I_k(D)$ denotes the $k$th Hodge ideal of $D$. The conjecture proposes continuity of Hodge ideals under sufficiently high $\mathfrak m_x$-adic approximation; the supplied text gives no resolution status.