Riemann–Hilbert correspondence for discrete Painlevé equations

The generalised Halphen surface and the corresponding Looijenga pair are the geometric objects associated with the discrete Painlevé equation under consideration. Riemann–Hilbert correspondence conjecture. For the elliptic and multiplicative/additive discrete Painlevé equations, the Riemann–Hilbert correspondence assigns to the generalised Halphen surface in Table 4 the corresponding Looijenga pair. This conjecture proposes a direct monodromy-manifold interpretation for discrete Painlevé equations, including the multiplicative cases where such a notion is otherwise missing. The source does not indicate whether the correspondence has been proved.