The filtration-saturation conjecture for the Grassmannian cohomology ring

Let $R^{k,\ell}$ be the graded cohomology ring of the Grassmannian, and let $R^{k,\ell,m}$ be the subalgebra generated by $e_1,\ldots,e_m$. Denote by $R^{k,\ell}_d$ and $R^{k,\ell,m}_d$ their degree-$d$ components.

The filtration-saturation conjecture. For $1\leq m\leq k$,

$$R^{k,\ell}_d=R^{k,\ell,m-1}_d\quad\text{for }d\geq k\ell-m^2+m+1.$$

The claim says that sufficiently high-degree classes are already generated by $e_1,\ldots,e_{m-1}$. The supplied source presents it as one of its conjectures and gives no resolution status.