The full Perazzo conjecture for minimal Artinian Gorenstein Hilbert functions

Let $m\geq 3$ and $d\geq 4$, and let $H$ be the Hilbert vector of a full Perazzo algebra of type $m$ and socle degree $d$. Write $r=r(m,d)$ for its codimension. Let $\mathcal{AG}(r,d)$ denote the family of standard graded Artinian Gorenstein algebras over a fixed field of characteristic $0$, of codimension $r$ and socle degree $d$, ordered by coordinatewise comparison of their symmetric Hilbert vectors. A Hilbert vector is minimal in this family if every comparable Artinian Gorenstein Hilbert vector below it is equal to it.

The full Perazzo conjecture. The Hilbert vector $H$ is minimal in $\mathcal{AG}(r,d)$; equivalently, if $\hat{H}\preceq H$ is a comparable Artinian Gorenstein Hilbert vector, then $\hat{H}=H$.

Full Perazzo algebras arise from full Perazzo polynomials through Macaulay–Matlis duality. The conjecture asserts that these algebras realize minimal Hilbert vectors for their codimension and socle degree; the supplied text gives no resolution.