The $e_m$-overkiller conjecture for the Grassmannian cohomology ring

Let $R^{k,\ell}$ be the graded cohomology ring of the Grassmannian, with generators $e_1,\ldots,e_k$, and let $R^{k,\ell,m-1}$ be the subalgebra generated by $e_1,\ldots,e_{m-1}$.

The $e_m$-overkiller conjecture. For $3\leq m\leq k$,

$$\ne_m e_1^{k\ell-2m}\in R^{k,\ell,m-1}.$$

The source notes that this stronger-looking product containment would imply the exact $e_m$-killer conjecture, and relates it to a weaker saturation statement. Its resolution status is not supplied.