Irreducibility conjecture for RSK-standard representation socles

Let $cmathfrak{m}$ be a multisegment, and let $cLambda(cmathfrak{m})$ be its RSK-standard representation. Let $Z(cmathfrak{m})$ denote the irreducible representation attached to $cmathfrak{m}$, and let $coperatorname{soc}$ denote the socle, the maximal semisimple subrepresentation. Socle irreducibility conjecture. The socle $coperatorname{soc}(cLambda(cmathfrak{m}))$ is irreducible, and

$$Z(\mathfrak{m})\cong\operatorname{soc}(\Lambda(\mathfrak{m})).$$

The authors cannot prove this in general; it would establish that the newly constructed RSK-standard representations have a unique irreducible subrepresentation, paralleling the corresponding property of traditional standard modules.