Extremal Betti-number conjecture for random higher-degree monomial ideals

Fix integers $d,m$ with $d\geq m\geq 1$, and suppose that $\alpha$ satisfies

$$\frac{1}{\binom{d+1}{m}}<\alpha<\frac{1}{\binom{d}{m}}.$$

Let $\Delta\sim\Delta_m(n,n^{-\alpha})$ be the corresponding multiparameter random simplicial complex, and let $I_\Delta$ be its Stanley–Reisner ideal. Extremal Betti-number conjecture. With high probability,

$$\mathbb{Q}[x_1,\ldots,x_n]/I_\Delta$$

has regularity and depth both equal to $d+1$, and has one extremal Betti number, namely $\beta_{n-(d+1),n}$. The conjecture proposes that random squarefree monomial ideals generated in fixed degree exhibit algebraic behavior analogous to random quadratic monomial ideals, but its validity is left open in the source.