Synchronization and large sets of Steiner systems in Johnson schemes

Let $J(n,k)$ be the Johnson scheme on the $k$-subsets of an $n$-set. A large set of Steiner systems is a collection of Steiner systems whose block sets partition all $k$-subsets. The Johnson scheme is non-synchronizing when the associated action fails to be synchronizing.

Large-set conjecture. There is a function $L$ such that, if $n\geqslant L(k)$, then the Johnson scheme $J(n,k)$ is non-synchronizing if and only if there exists a large set of Steiner triple systems $S(t,k,n)$ for some $t$ with $0<t<k$.

The conjecture links failure of synchronization to partitions of the $k$-subsets into Steiner-system block sets, while the preceding discussion relates such partitions to equality of clique and chromatic numbers.