Bombieri's conjecture for odd-even coefficient pairs below the line $n=(4m+2)/5$

Let $m>n\geq2$ be integers, with $m$ odd and $n$ even. Let $B_{mn}$ denote the trigonometric number

$$B_{mn}=\min_{t\in\mathbb R}\frac{n\sin t-\sin(nt)}{m\sin t-\sin(mt)}.$$

Bombieri's odd-even refinement. If

$$n<\frac{4m+2}{5},$$

then the equality between the Bombieri number $\sigma_{mn}$ and $B_{mn}$ is true.

This is presented as a proposition that should be true, based on numerical graph inspection, rather than as an established result. It proposes a strengthening of the range in which the universal Bombieri conjecture fails.