Algebraicity conjecture for cluster generating functions of generalized consecutive patterns

Let $m\geq4$ and let $\tau=134\ldots(m-1)2m$ be the consecutive pattern of length $m$. Let $T(x)$ be the cluster generating function specified by the preceding conjectural hypergeometric formula. Algebraicity conjecture. The solution $T(x)$ is algebraic, being the root of a polynomial of degree $m-2$. This conjecture extends the explicitly verified algebraic cases and predicts a uniform algebraic description for the cluster generating functions associated with this family of consecutive patterns; its validity for arbitrary $m\geq4$ remains open.