Complete homogeneous basis conjecture for Grassmannian cohomology subalgebras

Let $R^{\ell,k}$ be the cohomology ring of the Grassmannian, let $R^{\ell,k,m}$ be the ${\mathbb Q}$-subalgebra generated by elements of degree at most $m$, and let $\lambda^{\omega(k)}$ denote the $k$-conjugate of a $k$-bounded partition $\lambda$. Write $h_\lambda=h_{\lambda_1}h_{\lambda_2}\cdots h_{\lambda_r}$ for the complete homogeneous symmetric function indexed by $\lambda$.

Complete homogeneous basis conjecture. For $1\leq m\leq\min(\ell,k)$,

$$\{h_\lambda:\lambda_1\leq m\text{ and }\lambda^{\omega(k)}\subseteq(k^\ell)\}$$

is a ${\mathbb Q}$-basis of $R^{\ell,k,m}$. This conjecture would imply the Reiner–Tudose Hilbert-series conjecture by giving the required graded basis.