Higher Narayana number formula for local Weyl modules

Let $W_{\mathbb C^d}(\{0\}_{n\omega})$ be the indicated local Weyl module for $sl_2$, and let $W_{\mathbb C^d}(\{0\}_{n\omega})^{n-2i}$ denote its weight space of weight $n-2i$. Higher Narayana number conjecture. For $i=0,\dots,n$, one has

$$\dim W_{\mathbb C^d}(\{0\}_{n\omega})^{n-2i}=\frac{(d+n-1)!\cdots(d+n-i)!}{(n-1)!\cdots(n-i)!}\cdot\frac{2!\cdots(i-1)!}{d!\cdots(d+i-1)!},$$

and these dimensions equal the dimension of the irreducible representation of $sl_n$ with highest weight $d\omega_i$ for $i=1,\dots,n-1$, while they equal one for $i=0,n$. This proposes a higher-dimensional analogue of the Catalan and Narayana number formulas established earlier in the paper; the source gives no resolution of the proposed formula.