  In our previous work we have introduced an analogue of
Robinson-Schensted-Knuth correspondence for Schubert calculus of the complete
flag varieties. The objects inserted are certain biwords, the outcomes of
insertion are bumpless pipe dreams, and the recording objects are decorated
chains in Bruhat order. In this paper we study a class of biwords that have a
certain associativity property; we call them plactic biwords. We introduce
analogues of Knuth moves on plactic biwords, and prove that any two plactic
biwords with the same insertion bumpless pipe dream are connected by those
moves.
