Insertion algorithm realizing Kronecker coefficients

Construct an insertion algorithm that maps a bitableau \(B\) to a pair of tableaux \((T,T')\), preserves the weights \(w(T)=a(B)\) and \(w(T')=b(B)\), and ensures that for every triple of partitions \((\lambda,\mu,\nu)\), all pairs of tableaux of shapes \((\mu,\nu)\) have the same number of preimages among inputs of shape \(\lambda\).

Background

This problem reformulates the desired bicrystal construction entirely in terms of insertion. The input is a bitableau, while the output is a pair of tableaux whose shapes and weights encode the two representation-theoretic factors.

The equal-preimage condition would make the common number of preimages equal to the Kronecker coefficient, providing a combinatorial interpretation without explicitly referring to crystal operators.