Conjecture on linear-strand Betti numbers of balanced normal pseudomanifolds

Let $\Delta$ be a $(d-1)$-dimensional balanced normal pseudomanifold, with $d\geq 4$, and suppose that $f_0(\Delta)=kd$ for an integer $k\geq 2$. Let $\mathbb{F}[\Delta]$ denote its Stanley–Reisner ring, and let $\mathcal{ST}^{\times}(kd,d)$ denote the family of cross-polytopal stacked spheres on $kd$ vertices of dimension $d-1$. The linear-strand Betti number conjecture. For $\Gamma\in\mathcal{ST}^{\times}(kd,d)$ and every $i\geq 0$,

$$\beta_{i,i+1}(\mathbb{F}[\Delta])\leq\beta_{i,i+1}(\mathbb{F}[\Gamma]).$$

This conjecture is motivated by the fact that the computed Betti numbers of cross-polytopal stacked spheres are smaller than the previously established general bounds in the displayed example. The paper proves the relevant upper bounds and computes the cross-polytopal stacked-sphere Betti numbers, but leaves this sharper extremal comparison open.