Monotonicity conjecture for the expected beta-content function

Let $d$ and $n$ be as in the beta-polytope model, and fix parameters $\beta_1,\ldots,\beta_n\geq -1$. Write $\mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n}$ for the corresponding beta polytope, and let $\operatorname{Cont}_\beta$ denote its beta-content function. Monotonicity conjecture. The function

$$\beta\longmapsto \mathbb{E}\operatorname{Cont}_\beta(\mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n})$$

is increasing in $\beta$. The conjecture concerns the probability that a beta-distributed point is a vertex; the preceding discussion identifies the limiting cases $\beta=-1$ and $\beta\to+\infty$, but does not establish the asserted monotonicity.