Noble-number conjecture for the local minima of the Brjuno function

Let $B_1$ denote the classical Brjuno function, let $\mathcal{M}$ be the set of its local minima, and let $\mathcal{N}$ be the set of noble numbers, namely the numbers whose regular continued fraction expansion is of the form $[0;a_1,a_2,\ldots,a_n,\overline{1}]$. Local-minima conjecture.

$$\mathcal{M}=\mathcal{N}.$$

The global minimum of $B_1$ is known to be attained at the golden number, and the cited corollary gives the corresponding lower bound on $(0,1/2)$. Numerical evidence suggests that all local minima, not only these known global-minimum examples, are precisely the noble numbers.