Balanced algebraic-function conjecture for positive motherbody measures

Let $P(\mathcal C,z)$ be an irreducible polynomial, and call it balanced when $M(P)=0$. Suppose that the algebraic function defined by $P(\mathcal C,z)=0$ has a positive branch.

Balanced algebraic-function conjecture. Every such $P(\mathcal C,z)$ admits a positive motherbody measure.

This is the paper's main conjecture for balanced algebraic functions. The preceding discussion notes that the existence problem is substantially simpler in the balanced case than for polynomials with $M(P)>0$, but no general existence theorem is given.