Triangularity conjecture for RSK-standard representations

Let $\mathcal{M}$ be the set of multisegments. For a multisegment $cmathfrak{m}$, let $cLambda(cmathfrak{m})$ be its RSK-standard representation, let $Z(cmathfrak{m})$ be the corresponding irreducible representation, and let $cmathcal{RSK}(cmathfrak{m})$ be its RSK data. Write $[cLambda(cmathfrak{m})]$ and $[Z(cmathfrak{m})]$ for their classes in the Grothendieck group. The order $<$ is the product order induced by the domination order on inverted Young tableaux. Triangularity conjecture. For every multisegment $cmathfrak{m}$, there are multisegments $cmathfrak{n}_1,\dots,cmathfrak{n}_l$ such that

$$[\Lambda(\mathfrak{m})]=[Z(\mathfrak{m})]+\sum_{i=1}^l[Z(\mathfrak{n}_i)],$$

with $cmathcal{RSK}(cmathfrak{n}_i)<cmathcal{RSK}(cmathfrak{m})$ for every $i=1,\dots,l$. The multisegments $cmathfrak{n}_i$ need not be distinct, and not every smaller RSK datum need occur. The conjecture has been verified by computer calculation for all multisegments consisting of at most eight segments; the induced partial order is otherwise not known to have a geometric interpretation or simpler combinatorial description.