Commutativity conjecture for the tetrahedral X-operators

For a positive integer $N$, let $X_j^{(N)}(z)$, for $j=0,\dots,N$, be the tetrahedral $X$-operators depending on the spectral parameter $z$. Commutativity conjecture. For an arbitrary positive integer $N$, the operators satisfy

$$[X_j^{(N)}(z),X_j^{(N)}(w)]=0,\qquad j=0,\dots,N.$$

The identity is verified directly in the supplied text for $N=3$ when $j=0$ and $j=1$. A quantum-algebraic explanation for the commutativity in arbitrary positive rank is missing and remains an open problem.