Asymptotic sign coherence conjecture

Let $B=\hat B_{[N]}$ be irreducible. Let $\mu=(\mu_{k_j})_{j=1}^\infty$ be a sequence of matrix mutations applied to $\hat B$, and suppose that $\mu$ is monotone and balanced, meaning that its distance from the initial exchange matrix strictly increases at every step and every mutation direction occurs with positive lower asymptotic frequency. For a nonzero vector $\mathbf a\in\mathbb Z^N$, let $\sigma_\mu^{(n)}(\mathbf a)$ be the sign vector of the last row $\mathbf a^{(n)}$ after the first $n$ mutations.

Asymptotic sign coherence conjecture. There exists a sequence of strict sign vectors

$$\sigma_{\mathrm{reg}}=\left(\sigma_{\mathrm{reg}}^{(n)}\right)_{n=0}^\infty,\qquad \sigma_{\mathrm{reg}}^{(n)}\in\{+,-\}^N,$$

such that for every nonzero $\mathbf a\in\mathbb Z^N$ there is $T\in\mathbb N$ with

$$\sigma_\mu^{(n)}(\mathbf a)=\sigma_{\mathrm{reg}}^{(n)}$$

for all $n>T$.

The conjecture describes the eventual sign behavior of generalized $c$-vectors for arbitrary cluster algebras of geometric type. It is proved in the paper for rank-$2$ cluster algebras of infinite type and for a particular mutation sequence in the cluster algebra associated with the Markov quiver; the general assertion remains open.