The genus-three cluster-algebra description of the _2-quotient of Teichmller space

Let ${\mathcal T}_{3,0}$ be the Teichmuller space of flat $SL_2(\mathbb C)$-connections on a smooth Riemann surface of genus three. Consider the cluster algebra constructed in Section~ from the extended amalgamated quiver for $GL_4$. For each $\mathbb Z_2$-invariant geodesic $\gamma$, let $G_\gamma$ denote its geodesic function, and let $Z_+[a_1^{\pm 1/2},\dots,a_t^{\pm 1/2}]$ be the relevant positive Laurent polynomial algebra. For $M_i\in SL_2(\mathbb C)$, write $U_{i,j}=\operatorname{tr}(M_i^{-1}M_j)$. The genus-three cluster-algebra conjecture. This cluster algebra describes the $\mathbb Z_2$-quotient of ${\mathcal T}_{3,0}$ in such a way that:

• every $\mathbb Z_2$-invariant geodesic $\gamma$ corresponds to a geodesic function

$$G_\gamma\in Z_+[a_1^{\pm 1/2},\dots,a_t^{\pm 1/2}]$$

that is an element of an upper cluster algebra;

• all constructed $G_\gamma$ satisfy the skein relations and the Poisson Goldman bracket;
• to satisfy the rank condition, which is necessary for $U_{i,j}=\operatorname{tr}(M_i^{-1}M_j)$ with $M_i\in SL_2(\mathbb C)$, the Casimir must be set to $C=-1$.

The statement proposes a cluster-algebraic realization of the genus-three Teichmuller space quotient, encoding geodesic functions and their skein and Poisson structures. The supplied text does not indicate whether this assertion has been proved or remains open.