Higher level RSK correspondence

Let $\ell$ be a positive level. Denote by

$$\text{ParMat}^\flat=\bigcup_{m\ge 0}\bigcup_{\mu,\nu\in\Lambda_{\text{st}}^\ell(m)}\text{ParMat}^\flat_{\nu,\mu}$$

the set of bounded-partition-enhanced $\ell\times\ell$-block $\mathbb{N}$-matrices, and by

$$\text{SST}^2=\bigcup_{m\ge 0}\bigcup_{\mu,\nu\in\Lambda_{\text{st}}^\ell(m)}\text{SST}^2_{\nu,\mu}$$

the set of pairs of semistandard tableaux of the same $\ell$-multipartition shape. Higher level RSK correspondence. There exists a combinatorial construction of a one-to-one correspondence

$$\psi_\ell:\text{ParMat}^\flat\longrightarrow\text{SST}^{\,2}$$

(or, more explicitly, bijections $\psi_\ell:\text{ParMat}^\flat_{\nu,\mu}\longrightarrow\text{SST}^{\,2}_{\nu,\mu}$ for all $\mu,\nu\in\Lambda_{\text{st}}^\ell(m)$), such that $\psi_1$ is the RSK correspondence and $\psi_\ell$ extends $\psi_{\ell-1}$. The paper establishes the categorical higher-level correspondence $\varphi_\ell$, but a combinatorial construction with these properties is proposed as an object of interest and is not resolved here.