The exact $e_m$-killer 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 exact $e_m$-killer conjecture. For $1\leq m\leq k$,

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

This is a concrete consequence of the filtration-saturation claim: it asserts that the indicated product involving the next generator is already expressible using the preceding generators. The supplied source gives no resolution status.