The maximum-independent-set structure conjecture for qualitative independence graphs

Let $QI(k^2,k)$ be the qualitative independence graph, and for distinct $a,b\in\{1,\dots,k^2\}$ let $S_{\{a,b\}}$ denote the independent set defined in the paper.

Maximum-independent-set structure conjecture. For all positive integers $k$, every maximum independent set in $QI(k^2,k)$ is of the form $S_{\{a,b\}}$ for distinct $a,b\in\{1,\dots,k^2\}$.

This extends the corresponding result known for $QI(9,3)$. If true, the structural description would support proving that $QI(k^2,k)$ is a core for every $k$.