Convergence of the Sinkhorn orthogonalization algorithm

Let $U_N^N$, $Y_N$, $K_N$, and $X_N$ be the spaces and maps associated to the orthogonalization procedure, with maps $\Phi,\Psi$ and volume function

$$vol:U_N^N\to Y_N\to[0,1],\quad vol(u)=\prod_j\left|\det((u_{ij})_i)\right|.$$

Convergence conjecture. The maps $\Phi$ and $\Psi$ increase the volume and, after an infinite number of steps, respectively land in $K_N$ and $X_N$. This conjecture would imply convergence of the Sinkhorn-type iteration and provide an integration on $K_N$ and $X_N$ by push-forward of Haar measure.