Insertion algorithm for Kronecker coefficients

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

Background

The paper reformulates the crystal problem in terms of an insertion procedure, modeled conceptually on RSK and related algorithms. The desired map should retain the two bitableau weights as tableau weights and distribute inputs uniformly over pairs of output tableaux of prescribed shapes.

Such an algorithm would yield a combinatorial interpretation of $g(\lambda,\mu,\nu)$ by counting the occurrences of output pairs for inputs of shape $\lambda$. The authors explicitly state this as an open problem and note that it is weaker than the full crystal formulation.