The all-face-dimensions conjecture for block-beta random polytopes

Let $d=d_1+\cdots+d_m$, let $\bbbeta$ be the block-parameter vector, and let $\mathcal{P}_{d,d}^{\bbbeta}$ be the associated block-beta random polytope. All-face-dimensions conjecture. For every $j\in\\{0,1,\ldots,d-2\\}$,

$$\mathbb{E}f_j(\mathcal{P}_{d,d}^{\bbbeta})\asymp\mathbb{E}f_{d-1}(\mathcal{P}_{d,d}^{\bbbeta}).$$

The paper proves this comparison only in the range $j\geq\lfloor d/2\rfloor$ and conjectures that the restriction is unnecessary; the lower-dimensional face cases remain open.