Conjectural combinatorial formula for the parabolic H-triangle

For positive integers $m,n,t$, let $\operatorname{\bf NN}_{n,t}^{(m)}$ be the set of geometric multi-chains of $t$-filters. For $\mathcal V\in\operatorname{\bf NN}_{n,t}^{(m)}$, let $\operatorname{FL}(\mathcal V)$ denote its set of colored floors, let $S_{n,t}=\{(t,t+1),(t+1,t+2),\ldots,(n-1,n)\}$, and define

$$\tilde H_{n,t}^{(m)}(x,y)=\sum_{\mathcal V\in\operatorname{\bf NN}_{n,t}^{(m)}}x^{|\operatorname{FL}(\mathcal V)|}y^{|\operatorname{FL}(\mathcal V)\cap S_{n,t}|}.$$

Combinatorial $H$-triangle conjecture. For positive integers $m,n,t$,

$$\tilde H_{n,t}^{(m)}(x,y)=H_{n,t}^{(m)}(x,y)$$

with

$$\tilde H_{n,t}^{(m)}(x,y)=\sum_{k=0}^{n-t}\sum_{h=0}^{n-t-k}\left(\binom{mn-t+1}{k}\binom{t+k+h-2}{h}-m\binom{mn-t}{k-1}\binom{t+k+h-1}{h}\right)x^{n-t-k}y^{n-t-k-h}.$$

This asserts that the generating polynomial obtained from geometric multi-chains of $t$-filters equals the previously defined parabolic $H$-triangle. The supplied text does not state whether the identity has been proved or disproved.