Associated-prime conjecture for second powers of Kneser graph neighborhood ideals

Let $G=K(n,k)$ be the Kneser graph of $k$-element subsets of an $n$-element set, and let $NI(G)$ denote its closed neighborhood ideal in the corresponding polynomial ring with homogeneous maximal ideal $\mathfrak m$. Associated-prime conjecture. Then $\mathfrak m$ is an associated prime of $NI(G)^2$ if and only if $k\geq 2$ and $n=3k-1$. The proposition preceding this conjecture proves the assertion when $n\geq 3k-1$; the conjecture concerns the remaining range $k\geq 3$ and $2k+1\leq n\leq 3k-2$.