Goncharov–Shen's cluster Donaldson–Thomas transformation conjecture

Let $\mathcal{C}_{\mathfrak{g},\Sigma}$ be the mutation class associated to an admissible pair $(\Sigma,\mathfrak{g})$. Let $\mathbf{w}_0=(w_0,\ldots,w_0)\in W(\mathfrak{g})^p$ be the tuple of longest Weyl-group elements, let $\mathbf{r}_\Sigma$ be the mapping class rotating the special points on each boundary component of $\Sigma$ by one in the orientation induced by $\Sigma$, and let $*$ be the canonical Dynkin involution in $\operatorname{Out}(G)$. Goncharov–Shen's conjecture. For an admissible pair $(\Sigma,\mathfrak{g})$, the cluster Donaldson–Thomas transformation of $\mathcal{C}_{\mathfrak{g},\Sigma}$ is given by

$$\mathbf{r}_\Sigma\circ *\circ\mathbf{w}_0.$$

This was proved for $\mathfrak{g}=A_n$; the paper establishes a related implication for admissible disks with classical finite-type $\mathfrak{g}$.