Deformation conjecture for dessins d'enfants

A dessin d'enfant is a combinatorial object associated with a trigonal curve; it is maximal when it has one singular fiber in each region and is connected. Deformation conjecture. Every non-maximal dessin d'enfant can be deformed to a maximal dessin d'enfant. The preceding theorem establishes this deformation under the condition that every region with more than one $\times$-vertex satisfies the condition in Proposition; the conjecture asserts that no such restriction is necessary.