The all-degree Hard Lefschetz conjecture for matroid Gorenstein rings

Let $M=(E,\mathcal I)$ be a matroid of rank $d$, let $\A(M)$ be its associated Gorenstein ring, and let $a\in\mathbb R^E_{>0}$. All-degree Hard Lefschetz conjecture. The ring $\A(M)$ satisfies $\HL_k$ for some $a\in\mathbb R^E_{>0}$ for every $k\leq\frac d2$. The paper subsequently states that this conjecture is false, so the proposed assertion is refuted.