Abdulhadi–Hengartner conjecture on the valence of logharmonic polynomials

Let $f(z)=p(z)\overline{q(z)}$ be a logharmonic polynomial, where $p$ and $q$ are analytic polynomials of degrees $n$ and $m$, respectively, with $p$ not a constant multiple of $q$. Let the valence of $f$ be the number of preimages of a prescribed $w\in\mathbb{C}$. Abdulhadi–Hengartner conjecture. For every $m,n\geq 1$, the maximal valence is strictly less than the Bézout bound $(m+n)^2$. The paper proves the stronger upper bound $n^2+m^2$, so this conjecture is solved.