Sharp consecutive gap conjecture for two-point subsets of the circle

Let $(x_n)_{n\geq 1}$ be any sequence of distinct points on the circle $\mathbb T$. For each $n$, let $M_n^{(2)}$ and $m_n^{(2)}$ denote the quantities defined in the paper for two-point subsets. Sharp two-point gap conjecture.

$$\limsup_{n\to\infty}\frac{M_n^{(2)}}{m_n^{(2)}}\geq 2.$$

The proved general lower bound gives only $5/3$ when $r=2$, while numerical experiments suggest that the sharp value may be larger. The conjecture proposes the value $2$ as a strengthened lower bound.