The tricolour graph construction of four-critical-value rational functions

Let a tricolour graph be a graph of the type defined in the source, and let $z_1\in\mathbb{CP}^1$. Let $y$ denote the unique second-order critical point associated with the fourth critical value. A rational function of $z_1$ has four critical values, three normalized to $0$, $1$, and $\infty$, while its dependence on $y$ may be algebraic.

Tricolour graph construction. For any tricolour graph there exists a function

$$z=R(z_1,y)$$

which is rational in $z_1\in\mathbb{CP}^1$, has four critical values, three of them $0$, $1$, and $\infty$, and whose corresponding critical points are related to the graph as in the tricolour graph correspondence. The variable $y$ is the unique second-order critical point corresponding to the fourth critical value, and $R$ is an algebraic function of $y$ of genus zero.

This conjecture asserts the inverse construction from tricolour graphs to rational functions used in the paper's study of deformations and the sixth Painlevé equation. The supplied text does not state whether it has been proved or remains open.