Dipper–James standard basis conjecture for unipotent Specht modules

Let $\lambda\vdash n$. For each $\mathfrak s\in\operatorname{Std}(\lambda)$, let $r_{\mathfrak s}(t)\in\mathbb Z[t]$ and let $\mathcal B_{\mathfrak s}\subset S^\lambda$ be independent of $q$ and $F$. Dipper–James standard basis conjecture. These data satisfy

$$r_{\mathfrak s}(1)=1,$$ $$|\mathcal B_{\mathfrak s}|=r_{\mathfrak s}(q),$$

the union $\mathcal B=\mathcal B^\lambda=\bigcup_{\mathfrak s\in\operatorname{Std}(\lambda)}\mathcal B_{\mathfrak s}$ is disjoint, and $\mathcal B$ is a basis of $S^\lambda$. The polynomials $r_{\mathfrak s}(t)$ are called rank polynomials and $\mathcal B$ the standard basis. The paper gives a representation-theoretic proof in the special case $\lambda=(n-m,m)\vdash n$ with $0\leqslant m\leqslant n/2$; the unrestricted statement is not resolved in the supplied text.