Integrability conjecture for the cluster maps $\psi_{D_{2N}}$

Let $N\geq 3$, and let $\psi_{D_{2N}}$ denote the cluster map of type $D_{2N}$. Its iteration induces discrete dynamics, and the degree growth of the cluster map is quadratic, implying that the associated algebraic entropy vanishes.

Integrability conjecture. The discrete dynamics induced by the iteration of cluster maps $\psi_{D_{2N}}$ is a discrete integrable system.

The conjecture is motivated by the quadratic degree growth and vanishing algebraic entropy established in the surrounding discussion. The supplied text gives no resolution, so it remains open.