Uniqueness conjecture for the positive zeros of the twisted-crown function

Let $F$ be the function defined in the paper's equation for the twisted $(2,n)$-crowns, with $n\geq 4$. Its positive zeros are denoted by $z_1$ and $z_2$. Uniqueness conjecture. The function $F$ has only two positive solutions, denoted by $z_1$ and $z_2$. The authors report that numerical simulations suggest this uniqueness, but they have not proved it analytically; the question is also unresolved in related studies of twisted rings and regular polygons.