The filtered Hilbert-series conjecture for the Grassmannian cohomology ring

Let $R^{k,\ell}$ be the graded cohomology ring of the Grassmannian, with homogeneous generators $e_1,\ldots,e_k$, and let $R^{k,\ell,m}$ be the subalgebra generated by $e_1,\ldots,e_m$. Write $\operatorname{Hilb}(A,q)=\sum_{d\geq 0}\dim_{\mathbb Q}(A_d)q^d$ for the Hilbert series of a graded algebra $A$.

The filtered Hilbert-series conjecture. For $m\geq 0$,

$$\operatorname{Hilb}(R^{k,\ell,m},q)=1+\sum_{i=1}^m q^i\left[\begin{matrix}\ell\\i\end{matrix}\right]_q\left(\sum_{j=0}^{k-i}q^{j(\ell-i+1)}\left[\begin{matrix}i+j-1\\j\end{matrix}\right]_q\right).$$

This conjecture predicts the Hilbert series of each algebra in the natural filtration $\mathbb Q=R^{k,\ell,0}\subset\cdots\subset R^{k,\ell,k}=R^{k,\ell}$ and would describe how the cohomology ring is generated degree by degree. Its status is not resolved in the supplied source material.