Song–Wang's cyclotomic RSK bijection conjecture

For a level $\ell$ and multipartition data $\mu,\nu$, let $\mathrm{ParMat}_{\nu,\mu}^{\flat}$ denote the corresponding set of enriched block matrices, let $\mathrm{SST}_{\ell}(\lambda,\mu)$ denote the set of level-$\ell$ semistandard tableaux of shape $\lambda$ and content $\mu$, and let $\lambda \vdash_{\ell} n$ range over level-$\ell$ multipartitions of $n$. Song–Wang's conjecture. There exists, for each level $\ell$, a combinatorial bijection

$$\psi_\ell: \mathrm{ParMat}_{\nu,\mu}^{\flat} \longrightarrow \bigsqcup_{\lambda \vdash_\ell n} \mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu),$$

which extends the level $\ell-1$ bijection and recovers the classical RSK correspondence when $\ell=1$. This conjecture proposes a cyclotomic analogue of the classical RSK correspondence, relating the block-matrix basis data in the cyclotomic web category to pairs of tableaux in the cyclotomic Schur algebra; its status is not established in the supplied text.