The multiplicative Thom–Sebastiani conjecture for Bernstein–Sato roots of ideals

Let $\mathfrak a\subseteq\mathbb C[\bm x]$ and $\mathfrak b\subseteq\mathbb C[\bm y]$ be non-zero ideals. Let $W_{\mathfrak a}$ and $W_{\mathfrak b}$ denote the sets of roots of their Bernstein–Sato polynomials, without counting multiplicity, and let $\mathfrak a\mathfrak b$ be their product ideal.

Multiplicative Thom–Sebastiani conjecture. One has

$$W_{\mathfrak a}\cup W_{\mathfrak b}\subseteq W_{\mathfrak a\mathfrak b}.$$

Moreover,

$$W_{\mathfrak a}\cup W_{\mathfrak b}=W_{\mathfrak a\mathfrak b}\mod \mathbb Z.$$

The conjecture extends the inclusion and congruence modulo $\mathbb Z$ proved in the source for non-zero monomial ideals to arbitrary non-zero ideals. Its resolution status is not specified.