The negative-parameter extension conjecture for block-beta random polytopes

Let $\bbbeta=(\beta_1,\ldots,\beta_m)$ be the block-parameter vector for the block-beta random polytope $\mathcal{P}_{n,d}^{\bbbeta}$. Negative-parameter extension conjecture. The results of the paper's main theorem and its corollary hold for

$$\bbbeta\in(-1,\infty)^m.$$

The paper establishes the results for $\bbbeta\in[0,\infty)^m$; extending them to the full range where the beta distributions are defined is left open because the densities become unbounded and the beta-adjusted dimensions are no longer uniformly bounded above.