Monotonicity of the minimum odious-prime excess

Let $\mathbb{N}^e$ and $\mathbb{N}^o$ be the positive integers having, respectively, an even and an odd number of $1$'s in their binary expansions. Let $\mathbb{P}^e$ and $\mathbb{P}^o$ be the corresponding sets of evil and odious primes, and let $\pi^e(x)$ and $\pi^o(x)$ count these primes not exceeding $x$. Define

$$m_n=\min\limits_{x\in(2^{n-1},2^n)}(\pi^o(x)-\pi^e(x)).$$

Monotonicity conjecture. For all $n\in\mathbb{N}$, $n\neq 5,6$, the sequence $m_n$ increases monotonically.

This conjecture formulates the observed numerical phenomenon that the odious primes are generally in excess of the evil primes, with the minimum excess over each dyadic interval increasing with $n$. The supplied statement is truncated after the exceptional values $n=5,6$, so the precise intended inequality should be checked against the original source.