The phase-transition conjecture for minima of $\sigma$-Brjuno functions

Let $B_\sigma:[0,1]\to\mathbb R$ be the $\sigma$-Brjuno function, and let $\eta_n=(\sqrt{n^2+4}-n)/2$ be the $n$-th fixed point of the Gauss map. For each $n\in\mathbb N$, consider the minimum of $B_\sigma$ on $[0,1]$. Phase-transition conjecture. For all $n\in\mathbb N$, there exists a value $\sigma_n^*\in(n-1,n)$ such that

$$\min_{x\in[0,1]}B_\sigma(x)= \begin{cases} B_\sigma(\eta_n)&\text{if }\sigma\in[n-1,\sigma_n^*)\\ B_\sigma(\eta_{n+1})&\text{if }\sigma\in[\sigma_n^*,n]. \end{cases}$$

Numerical evidence suggests that the minimizer remains locked at successive fixed points of the Gauss map and jumps between them at the critical thresholds $\sigma_n^*$; proving this description and the existence of these thresholds remains open.