Banica's convergence conjecture for the quantum Sinkhorn algorithm

For $n$ a positive integer, let the alternating normalization procedure be the row and column normalization algorithm in Algorithm, initialized by a random complex Gaussian tensor and applied to produce a quantum permutation matrix. For a precision parameter $\varepsilon>0$, the procedure tests whether the maximal row-sum error is below $\varepsilon$. Banica's convergence conjecture. For any precision parameter $\varepsilon>0$ and for almost all initializations of the alternating normalization procedure, the program will terminate successfully after a finite number of steps. The conjecture concerns almost-sure finite termination of the proposed algorithm; convergence is known when unit rank matrices are replaced by positive definite elements, but the stated quantum permutation-group setting remains unresolved.