Skew RSK and the switching on ballot tableau pairs

In arXiv:1808.06095 we have introduced the Knuth class of the word recording a sequence of locations for repeated internal insertion operations in the Sagan-Stanley skew RSK correspondence, with no prescribed external insertion of new cells, to be a preserver for the $P$-tableau. As a consequence the Benkart-Sottile-Stroomer switching involution on ballot tableau pairs allows a realization as a recursive internal insertion procedure. This amounts to explain the various presentations of Littlewood-Richardson (LR) commuters and their coincidence predicted by Pak and Vallejo with contributions by Danilov and Koshevoi. In particular, the aforesaid presentation provides internal insertion as an alternative to Sch\"utzenberger- Lusztig involution (or evacuation) to constructing the Gelfand-Tsetlin pair in the Henriques-Kamnitzer $\mathfrak{gl}_n$-crystal commuter. In addition, the coincidence of LR commuters solves the Lecouvey-Lenart conjecture, recently further developed by Kumar-Torres, on bijections between the Kwon and Sundaram branching models.