This Lemma is parallel to Lemma 3.10, so we will only highlight the key issues. The constants $\theta_{13}^{\infty}$ and $\theta_{23}^{\infty}$ are the asymptotic holonomy as $\mu\to+\infty$ of the $S^{1}$-connection $\vartheta$, along the $S^{1}$-cycles in the base corresponding to the $x_{1}$ and $x_{2}$ variables respectively. These are introduced in order to cancel the twist of $\vartheta$ by a flat connection.

The functional equation follows from

which crucially uses Lemma 4.25. ∎