Exact top-dimensional homology rank conjecture for independence complexes of Kneser graphs

Let $n\geq 3$, $k\geq 1$, and let $\operatorname{Ind}(\operatorname{KG}(n,k))$ denote the independence complex of the Kneser graph. Set

$$p=\frac{1}{2}\binom{2n}{n}-1.$$

Exact homology-rank conjecture. The rank of the $p$-dimensional homology group of $\operatorname{Ind}(\operatorname{KG}(n,k))$ is

$$\binom{2n+k}{2n}.$$

The conjecture is motivated by computed results for $n=3$ and $k=1,2,3$, where the lower bound proved earlier in the paper is attained. Its validity beyond those computations remains open.