Meagher–Purdy conjecture on intersecting families of multisets

Let $k,n,t$ be positive integers with $t\leq k$, let $M(n,k)$ be the rectangle representation of $k$-multisets, and let $\mathcal{M}(n,k,t)$ denote the families of $t$-intersecting $k$-multisets of $[n]$. Meagher–Purdy conjecture. If

$$t(k-t)+2\leq n$$

and $\mathcal{F}\in\mathcal{M}(n,k,t)$, then

$$|\mathcal{F}|\leq\binom{n+k-t-1}{k-t}.$$

Moreover, if $n>t(k-t)+2$, equality holds if and only if every member of $\mathcal{F}$ contains a fixed $t$-multiset of $M(n,k)$. This gives the paper’s proposed explicit threshold for the multiset Erdős–Ko–Rado bound; the paper’s abstract states that the numerical bound is established for this threshold, while the full equality assertion is the conjectural part recorded here.