The t-intersecting multiset conjecture

Let $[n]=\{1,\ldots,n\}$, and let a $k$-multiset on $[n]$ be a multiset of cardinality $k$ whose elements lie in $[n]$. Two multisets are $t$-intersecting when their intersection has size at least $t$, counted with multiplicity. Let $n\geq n_0(k,t)$. The $t$-intersecting multiset conjecture. If $\mathcal{M}$ is a $t$-intersecting collection of $k$-multisets on $[n]$, then

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

Furthermore, equality is achieved if and only if $\mathcal{M}$ is a trivial collection. This proposes the corresponding Erdős–Ko–Rado-type extremal bound for multisets; the source presents it as an additional direction and gives no resolution.