Partial-link Singer conjecture for aspherical polyhedra

Let $X$ be an $n$-dimensional aspherical polyhedron. Say that $X$ has spherical links in codimensions at most $m$ if, for each $i\le m$, the link of every $(n-i)$-cell is an $(i-1)$-sphere. Let $\widetilde X$ be the universal cover.

Partial-link Singer conjecture. If $X$ has spherical links in codimensions at most $2l+1$, where $2l+1\le n$, then

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

The case $l=0$, corresponding to pseudomanifolds, holds for elementary reasons. The conjecture is motivated by the inductive program in the paper and is otherwise open in general.