RSK-standard basis conjecture for the Grothendieck subgroup

Let $cmathfrak{M}$ be the set of multisegments, let $cLambda(cmathfrak{m})$ be the RSK-standard representation attached to $cmathfrak{m}cincmathfrak{M}$, and let $cmathcal{R}'$ be the subgroup of the direct sum of the Grothendieck groups of the representations of $cmathrm{GL}_n(F)$ generated by the classes $[Z(cmathfrak{m})]$. RSK-standard basis conjecture. The classes

$$[\Lambda(\mathfrak{m})],\qquad \mathfrak{m}\in\mathfrak{M},$$

form a $cmathbb{Z}$-basis of $cmathcal{R}'$. This is stated as a weaker consequence of the triangularity conjecture; the paper does not establish it in general.