The strong full Perazzo conjecture for strongly minimal Hilbert functions

Let $m$ and $d$ be the type and socle degree of a full Perazzo algebra, let $r=r(m,d)$ be its codimension, and let $H$ be its Hilbert vector. For $k\leq d/2$, define

$$\mu_k(r)=\min\{h_k\mid h_k=\dim A_k\text{ for }A\in\mathcal{AG}(r,d)\},\qquad \delta_k(r)=r-\mu_k(r).$$

Here $\mathcal{AG}(r,d)$ is the family of standard graded Artinian Gorenstein algebras of codimension $r$ and socle degree $d$, ordered by comparison of their symmetric Hilbert vectors. The vector $H$ is strongly minimal if it is realized by an Artinian Gorenstein algebra, is minimal in $\mathcal{AG}(r,d)$, satisfies $h_k=\mu_k(r)$ for every $k\in\{2,3,\ldots,\lfloor d/2\rfloor\}$, and satisfies $\delta_k(r-1)<\delta_k(r)$ for every such $k$.

The strong full Perazzo conjecture. The Hilbert vector $H$ is strongly minimal in $\mathcal{AG}(r,d)$.

This is presented as a stronger conjecture than the full Perazzo conjecture, imposing degreewise minimality and strict growth conditions in addition to poset minimality. The supplied text gives no resolution.