The indecomposability criterion for the $n/d$-conjecture

Let $f\in\operatorname{Sym}^{d}((\mathbb{C}^{n})^{*})$ be a homogeneous polynomial of degree $d$ in $n$ variables. For a decomposition $(\mathbb{C}^{n})^{*}=W_{1}\oplus W_{2}$ and an integer $k$ with $0\leq k\leq d$, consider the natural multiplication map

$$\operatorname{Sym}^{k}(W_{1})\otimes\operatorname{Sym}^{d-k}(W_{2})\longrightarrow\operatorname{Sym}^{d}((\mathbb{C}^{n})^{*}).$$

The indecomposability criterion for the $n/d$-conjecture. If, for every nontrivial decomposition $(\mathbb{C}^{n})^{*}=W_{1}\oplus W_{2}$ and every integer $0\leq k\leq d$ satisfying $nk\neq d\,\dim W_{1}$, the polynomial $f$ does not belong to the image of this map, then $-\frac{n}{d}$ is a root of the local Bernstein–Sato polynomial $b_{f,0}(s)$.

This proposes a representation-theoretic condition under which the distinguished value $-n/d$ occurs as a Bernstein–Sato root. The source presents it as a new conjecture, and no resolution is supplied.