Top-degree and socle-dimension conjecture for Solomon–Terao algebras

Let ${\mathcal{A}}$ be a hyperplane arrangement, let $\eta\in U_d({\mathcal{A}})$, and define

$$r:=\max\{n\mid ST({\mathcal{A}},\eta)_n\neq(0)\}.$$

Top-degree and socle-dimension conjecture. The top nonzero degree and its dimension satisfy

$$r=|{\mathcal{A}}|+\ell(d-2),$$

and

$$\dim_{\mathbb{K}}ST({\mathcal{A}},\eta)_r=1.$$

This conjecture predicts a uniform top degree and one-dimensional top graded piece for Solomon–Terao algebras. It is presented as a question suggested by computations; no resolution is given in the source.