Radial quadratic-differential link-pattern conjecture

Let $\mathcal{QD}(\boldsymbol{z})$ be the specified space of quadratic differentials with $m$ poles and $n$ zeros, and let $\Gamma(Q)$ denote the horizontal trajectories of $Q(z)\,dz^2$. Radial quadratic-differential conjecture. Up to multiplication by a real constant: if $m\leq n/2$, $\Gamma(Q)$ forms an $(n,m)$-link and every link pattern is realized by a unique $Q\in\mathcal{QD}(\boldsymbol{z})$; if $(n+1)/2\leq m\leq n$, $\Gamma(Q)$ forms an $(n,n-m)$-link and every link pattern is realized by a continuous family of such differentials with the same real horizontal trajectories; and if $m>n$, no such $Q$ exists. This conjecture relates the classification of quadratic differentials to critical points of the trigonometric KZ equations; its resolution is not given in the paper.