Divisibility characterization of non-separation in Johnson schemes

Let $J(n,k)$ be the Johnson scheme on the $k$-subsets of an $n$-set. For a Steiner system $S(t,k,n)$, the divisibility conditions are

$$\binom{k-i}{t-i}\mid\binom{n-i}{t-i}\qquad(0\leqslant i\leqslant t-1).$$

Divisibility conjecture. There is a function $H$ such that, if $n\geqslant H(k)$, then the Johnson scheme is non-separating if and only if the divisibility conditions for $S(t,k,n)$ are satisfied for some $t$ with $0<t<k$.

This is the reformulation obtained by combining the preceding conjecture with Keevash's theorem, which gives Steiner systems for sufficiently large parameters satisfying the divisibility conditions.