Riemann–Hilbert uniqueness conjecture for Gaiotto–Moore–Neitzke coordinates

Let $\mathcal{Y}^{RH}_{\gamma}(\xi)$ be the piecewise holomorphic function on the $\xi$-plane obtained from the Fock–Goncharov coordinate data, with discontinuities on BPS $\mathcal{K}$-rays and the prescribed asymptotic behavior. Let equation (34) denote the integral equation for the functions $\mathcal{Y}_{\gamma}(\xi)$, and let $\mathcal{Y}$ be replaced there by $\mathcal{Y}^{RH}$. Riemann–Hilbert uniqueness conjecture. $\mathcal{Y}^{RH}_{\gamma}(\xi)$ satisfies equation (34) after replacing $\mathcal{Y}$ by $\mathcal{Y}^{RH}$. The source explains that this essentially asserts uniqueness of the solution to the Riemann–Hilbert problem and marks the conjecture as resolved.