Brenti's unimodality conjecture for standard graded Gorenstein domains

Let $A$ be a standard graded Gorenstein integral domain, and write its Hilbert series as $H(A;z)=h(A;z)/(1-z)^{\dim(A)}$, where $h(A;z)$ is the $h$-polynomial. Brenti's conjecture. The polynomial $h(A;z)$ is unimodal. This is a long-standing algebraic unimodality problem, inspired by a conjecture of Stanley; its status is not resolved in the supplied material.