Existence of deformation dessins for rational functions with three critical values

Let $R$ be a rational function in $z_1$ with deformation dimension $m=1$, satisfying the conditions of Proposition~[necessary condition], and suppose that its first three critical values are $0$, $1$, and $\infty$. A tricolour graph is a connected graph on the Riemann sphere with black, white, and one blue vertex, satisfying the stated incidence, cycle, face-boundary, and valency conditions. Existence conjecture. There exists a tricolour graph whose faces, white vertices, and black vertices correspond one-to-one with the critical points of $R$ having critical values $0$, $1$, and $\infty$, respectively, and whose black face orders and vertex valencies equal the multiplicities of the corresponding critical points. This asserts the dessin representation needed for the deformation theory of rational functions and its applications to the sixth Painlevé equation; the supplied text gives no evidence that the claim has been proved or refuted.