Non-separation and Steiner systems in Johnson schemes

Let $J(n,k)$ be the Johnson scheme on the $k$-subsets of an $n$-set. A Steiner system $S(t,k,n)$, for $0<t<k<n$, is a collection of $k$-subsets such that every $t$-subset is contained in exactly one block. The Johnson scheme is non-separating when its clique number times its coclique number equals its number of vertices.

Non-separation conjecture. There is a function $G$ such that, if $n\geqslant G(k)$ and the Johnson scheme $J(n,k)$ is non-separating, then there exists a Steiner system $S(t,k,n)$ for some $t$ with $0<t<k$.

A Steiner system yields equality between the clique–coclique product and the number of vertices, so the conjecture proposes an asymptotic converse. Together with Keevash's existence theorem for sufficiently large admissible parameters, it would characterize non-separation through the divisibility conditions for Steiner systems.