Steiner-system extremal conjecture for constant-column-weight disjunct matrices

Let $M$ be an $n\times m$ $t$-disjunct matrix with constant column weight $b$. Set

$$p=\frac{b+t-1}{t},$$

and assume that $p$ is an integer. A Steiner system $\mathfrak{S}(n,b,p)$ is a family of $b$-subsets of an $n$-element set such that every $p$-subset is contained in exactly one member; the maximum such family is denoted by $\mathfrak{S}(n,b,p)$. Steiner-system conjecture. The maximum $\mathfrak{S}(n,b,p)$ Steiner system gives a matrix $M'$ that is no worse than $M$, equivalently,

$$|\mathfrak{S}(n,b,p)|\ge m.$$

The conjecture proposes that Steiner-system constructions are at least as efficient as arbitrary constant-column-weight $t$-disjunct matrices with the same parameters. The surrounding discussion notes that the Steiner construction yields a $\left\lfloor\frac{b-1}{p-1}\right\rfloor$-disjunct matrix, but the supplied text gives no resolution of this conjecture.