Cohomological distinction conjecture for the groups $G(\bar n)$

Fix, for every integer $n\geq 1$, the fundamental group $G_n$ of an $(n+2)$-dimensional cusped hyperbolic manifold. Let $\bar n$ be a finite collection of integers at least $1$, and let $G(\bar n)$ be a finitely generated infinite group containing copies of all $G_n$ with $n\in\bar n$ as hyperbolically embedded subgroups. Write $\partial_*G(\bar n)$ for its Morse boundary. Cohomological distinction conjecture. The Čech cohomology satisfies

$$\check{H}^i(\partial_* G(\bar n),\mathbb{R})$$

non-trivial for $i\in\bar n$ and trivial for $i\notin\bar n$, with $1<i\leq\max(\bar n)$. The conjecture is intended to distinguish the quasi-isometry classes of the various groups $G(\bar n)$: the Morse boundary should contain copies of the boundaries $\partial_*G_n$, contributing cohomology in the appropriate degrees, while otherwise being $1$-dimensional.