Enumeration conjecture for critical points of the radial master function

Let $\boldsymbol{z}$ be generic on the unit circle, let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function, and write $\#\operatorname{LP}(n,m)$ for the number of $(n,m)$ link patterns. Critical-point enumeration conjecture. If $m\leq n/2$, then $\Phi_{m,n}$ has exactly $\#\operatorname{LP}(n,m)$ isolated critical points. If $(n+1)/2\leq m\leq n$, then, in the symmetric coordinates $\lambda_1,\ldots,\lambda_m$ on $\mathbb{C}^m_{\lambda}$, its critical points consist of $\#\operatorname{LP}(n,n-m)$ straight lines. If $m>n$, it has no critical points. The conjecture is the proposed critical-point counterpart of the quadratic-differential classification and remains open.