Conjectural derivative identity for moments of Minkowski's question mark function

Let $m_x$ be the holomorphic extension of the moments of Minkowski's question mark function, so that $m_n$ denotes its value at each positive integer $n$. The experimentally observed derivative at the origin is related to these moments by the following identity.

Conjectural derivative identity.

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

This identity is motivated by numerical agreement between the derivative of $m_x$ at $x=0$ and the corresponding expression involving the moments, but it is presented only as a conjecture in the source.