Mirror expansion formula for the logarithmic period

Let $z:=\phi^3$, let $\theta:=z\frac{d}{dz}$, and let $I_3$ be the logarithmic solution of the Picard–Fuchs equation described in the source. Define

$$q=-ze^{I^{(0)}_2}=e^{2\pi i\tilde I_2/\tilde I_1},$$

where $I^{(0)}_2$ is the holomorphic part of the second solution and $\tilde I_1,\tilde I_2$ are the indicated monodromy periods. Mirror expansion formula. The period $I_3$ satisfies

$$I_3=\frac{(\log(-q))^2}{2}+\sum_{d=1}^{\infty}(-1)^d(3d)^2m_d\sum_{k=1}^{\infty}\frac{q^{dk}}{k^2}.$$

This proposal matches the calculated $q$-expansion with the previously tabulated values of $m_d$, expressing the logarithmic period through the enumerative invariants. The source gives no proof or resolution of the formula.