Alternating conjectural relation for moments of Minkowski's question mark function

Let $m_x$ be the holomorphic extension of the moments of Minkowski's question mark function, and let $m_n$ denote its value at the positive integer $n$. A variation of the preceding experimentally motivated relation is proposed.

Alternating moment identity.

$$\log 2=\sum_{n=1}^\infty \frac{m_n}{n}\left(\frac{1}{2^n}-(-1)^n\right).$$

The source presents this as a variation of the preceding conjectural identity; no proof or resolution is supplied there.