Enumeration conjecture for geometric multi-chains of t-filters

For $n\geq 1$, let $T_n=\{(i,j)\mid 1\leq i<j\leq n\}$ be the triangular poset with $(i,j)\preceq(k,l)$ if and only if $i\geq k$ and $j\leq l$. For $t\in[n]$, let $T_{n,t}=\{(i,j)\in T_n\mid j>t\}$, and let $(V_m,V_{m-1},\ldots,V_1)$ be a geometric multi-chain of $t$-filters, meaning a nested tuple of filters satisfying the two geometricity conditions defined for the triangular poset.

Enumeration conjecture. The number of geometric multi-chains of $t$-filters $(V_m,V_{m-1},\ldots,V_1)$ is

$$\frac{mt+1}{mn+1}\binom{(m+1)n-t}{n-t}.$$

This conjectural formula gives a closed enumeration of the geometric multi-chains used in the proposed combinatorial description of the $H$-triangle. The supplied text does not state whether the formula has been proved or disproved.