Maximum cocliques in the projective-plane Johnson graph

Let $q>2$ and consider the graph $\Gamma_q(q^2+q+1,q+1)$ on the $(q+1)$-subsets of a $(q^2+q+1)$-set, with adjacency determined by the Johnson-scheme relation indexed by $q$. A maximum coclique is a coclique of largest possible cardinality.

Projective-plane coclique conjecture. For $q>2$, a coclique of maximum size in the graph $\Gamma_q(q^2+q+1,q+1)$ must consist of all the $(q+1)$-sets containing two given points; so the chromatic number of this graph is strictly larger than $q^2+q+1$.

The paper reports that the conjecture holds for $q=3$ and $q=4$. It concerns the relationship between projective planes, extremal cocliques, and synchronization versus separation in Johnson schemes.