Singer-type vanishing conjecture for flag complexes with spherical links

Let $L$ be an $(n-1)$-dimensional flag complex with spherical links in codimensions at most $2l+1$, where $2l+1\le n$. When $n=2l+1$, interpret this as saying that $L$ is an $(n-1)$-sphere. Let $\Sigma_L$ be the associated cubical complex.

Flag-complex partial-link conjecture.

$${\operatorname{\mathcal H}}_{n-i}(\Sigma_L)=0\quad\text{for }i\le l.$$

This is the right-angled Coxeter form of the partial-link Singer conjecture. The paper derives it in cases from the established low-dimensional instance, but the full statement remains open.