Fixed-point computation conjecture for harmonic-map polygons

Let $P$ be a polynomial with only simple zeroes, and let $Y(P)$ be the convex polygon associated to the minimal harmonic map determined by $P$. Let $\zeta$ be the variable on which the relevant collections of functions are defined, and let the integral operator act on these collections. Fixed-point computation conjecture. The polygon $Y(P)$ can be computed from the solution of a fixed-point problem for this integral operator. This is the paper's central conjectural method for determining the asymptotic polygon associated to a polynomial cubic differential; the source does not state a resolution.