Iterated socle conjecture for RSK-standard representations

Let $cmathfrak{m}$ be a multisegment with RSK data

$$\mathcal{RSK}(\mathfrak{m})=(\mathfrak{l}_1,\dots,\mathfrak{l}_k).$$

For each ladder $cmathfrak{l}_i$, let $Z(cmathfrak{l}_i)$ be its irreducible representation, and define representations recursively by $cpi'_1=Z(cmathfrak{l}_1)$ and $cpi'_i=\operatorname{soc}(Z(\mathfrak{l}_i)\times\pi'_{i-1})$ for $i=2,\dots,k$. Iterated socle conjecture.

$$Z(\mathfrak{m})\cong\pi'_k.$$

This is presented as a weaker form of the general socle-irreducibility conjecture and concerns the successive socles in the prescribed RSK ladder order.