The smallest-eigenspace dimension conjecture for qualitative independence graphs

For a positive integer $k$, let $QI(k^2,k)$ be the qualitative independence graph, and let $\frac{-(k!)^{k-1}}{k}$ be its smallest eigenvalue.

Smallest-eigenspace dimension conjecture. For all positive integers $k$, the eigenspace corresponding to this smallest eigenvalue has dimension

$$\binom{k^2}{2}-\binom{k^2}{1}.$$

The dimension is known for $k=3$, and the paper reports confirmation for $k=4$. The conjecture is motivated by the structure of maximum independent sets and possible proofs that these graphs are cores.