Kontsevich's noncommutative Laurent phenomenon conjecture

Recall that the Kontsevich map $K_r$, for $r\in\mathbb{Z}_{>0}$, is the birational automorphism of the noncommutative plane

$$K_r:(x,y)\mapsto (xyx^{-1},(1+y^r)x^{-1}).$$

For $r_1,r_2\in\mathbb{Z}_{>0}$, consider the iterations formed by alternating $K_{r_1}$ and $K_{r_2}$:

$$\underbrace{\cdots K_{r_1}K_{r_2}K_{r_1}}_k(x,y),\qquad k\geq 1.$$

Kontsevich's conjecture. For any $r_1,r_2\in\mathbb{Z}_{>0}$, all these iterations are given by noncommutative Laurent polynomials in $x$ and $y$.

The conjecture asserts a noncommutative analogue of the Laurent phenomenon for cluster-type birational transformations. The paper's stated aim is to give an elementary proof, but the supplied status is unknown, so its resolution is not determined from the provided material.