Irreducibility conjecture for Santaló varieties of generic polytopes

Let $A \in \mathbb{R}^{d \times n}_{\geq 0}$ be a generic matrix, let $C \in \mathcal{C}_A$ be a cell, and let $n_C$ denote the number of vertices associated with $C$. For $u \in \mathbb{C}^{n_C+1}$, write $\mathscr{X}_{C,u}$ for the corresponding Santaló variety, and let $\mathbf{1}$ be the all-ones vector.

Irreducibility conjecture. For generic matrices $A \in \mathbb{R}^{d \times n}_{\geq 0}$ and each cell $C \in \mathcal{C}_A$, there exists a dense open subset $U \subset \mathbb{C}^{n_C+1}$ such that $\mathscr{X}_{C,u}$ is irreducible of dimension $d$ for $u \in U$. Moreover, $\mathbf{1} \in U$ and

$$\mathscr{X}_{C,\mathbf{1}}=\mathcal{X}_C=X_C.$$

This conjecture would identify the generic parametrized Santaló variety with the varieties obtained from the unweighted and weighted constructions, and supports the degree comparisons used in the paper. No resolution is given in the source.