Product-of-triangles conjecture for irreducible Ferrers-diagram polytopes

For each integer $d\geq 3$, let $\mathfrak P_d\subseteq\mathbb R^{2d-3}$ be the integral polytope whose integer points correspond to irreducible Ferrers-diagram pairs $(\mathcal D,d)$. A triangle means a 2-simplex. Product-of-triangles conjecture. For all $d\geq 3$, the polytope $\mathfrak P_d$ is combinatorially equivalent to the Cartesian product of $d-2$ triangles. This has been verified computationally for $3\leq d\leq 7$, while the assertion for all $d$ remains open.