A note on a cyclotomic-friendly application of RSK

We give a combinatorial realization of a level-$\ell$ Robinson-Schensted-Knuth correspondence conjectured to exist by Song and Wang for cyclotomic Schur categories. We show that cyclotomic basis elements can be canonically reorganized into flagged block composition matrices encoding families of biwords, so that the correspondence is obtained by applying the classical RSK correspondence componentwise. This perspective identifies the level-$\ell$ correspondence as an iteration of classical RSK, specializing to the usual correspondence when $\ell=1$ and behaving naturally under restriction to lower levels.