The generalized eigenspace-dimension conjecture for uniform qualitative independence graphs

Let $UQI(ck,k)$ be the uniform qualitative independence graph on uniform $k$-partitions of a $ck$-set, and consider the eigenspace corresponding to its smallest eigenvalue.

Generalized eigenspace-dimension conjecture. For all positive integers $c,k$, this eigenspace has dimension

$$\binom{ck}{2}-\binom{ck}{1}.$$

The paper reports that the formula holds for the examples $QI(3c,3)$ with $c=3,4,5,6$, and suggests extending the earlier conjecture from $QI(k^2,k)$ to all $UQI(ck,k)$.