Equality criterion for Hodge ideals and microlocal Hodge ideals

Let $D=\alpha Z$, where $0<\alpha\le 1$ and $Z=(f=0)$ is an integral and reduced effective divisor defined by a weighted homogeneous polynomial $f$ with an isolated singularity at the origin. Let $I_k(D)$ and $\tilde{I}_k(D)$ be the Hodge ideals and microlocal Hodge ideals associated with $D$. Equality criterion. For every $k\in\mathbb{N}$,

$$I_{k+1}(D)=\tilde{I}_{k+1}(D)$$

if and only if

$$I_k(D)=\tilde{I}_k(D)=(x_1,\ldots,x_n)^m$$

for some $m\in\mathbb{N}$. This proposed equivalence extends the preceding one-way implication and the converse established in the case $k=0$; its validity beyond those cases is left open.