Weiner's enumeration conjecture for pattern-avoiding Grassmannian permutations

Let $\mathscr{G}_m(12\cdots k)$ denote the set of Grassmannian permutations of size $m$ that avoid the increasing pattern $12\cdots k$, and let $C_r=\frac{1}{r+1}\binom{2r}{r}$ be the $r$th Catalan number. For integers $k\geq 2$ and $m\in\{k,\dots,2k-2\}$, Weiner's conjecture.

$$\left|\mathscr{G}_{m}(12\cdots k)\right|=\sum_{j=1}^{k-\lfloor m/2\rfloor}(-1)^{j-1}j\binom{2k-m-j}{j}C_{k-j}.$$

The formula generalizes the previously established Catalan-number values at $m=2k-3$ and $m=2k-2$. It was verified by the authors for $k\leq 12$, but a proof for all $k$ and the stated range of $m$ remains open.