The plectic polylogarithm specialization conjecture for Lerch zeta values

Let $F$ be a totally real field of degree $g$, let $U$ be the relevant open torus, let $\boldsymbol{\operatorname{pol}}$ be the polylogarithm class, and let $\xi$ be a torsion point of $U$. Write $\Delta=\mathcal O_{F+}^{\times}$ and let $\xi\Delta$ be the orbit of $\xi$. Under the conjectural equivariant plectic Hodge theory and the resulting specialization map

$$i_{\xi\Delta}^*:H^{2g-1}_{\sD}(U/F^\times_+,\bbL\\!\operatorname{og})\longrightarrow H^{2g-1}_{\sD^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\bbL\\!\operatorname{og}),$$

with

$$H^{2g-1}_{\sD^I}(\xi\Delta/\Delta,i_{\xi\Delta}^*\bbL\\!\operatorname{og})=\prod_{n=1}^{\infty}(2\pi i)^{(n-1)g}\bbR,$$

Plectic polylogarithm specialization conjecture. The specialization satisfies

$$i_{\xi\Delta}^*\boldsymbol{\operatorname{pol}}=(d_F^{1/2}\cL^\infty(\xi\Delta,n))_{n=1}^{\infty}$$

where $d_F$ is the discriminant of $F$ and

$$\cL^\infty(\xi\Delta,n)=\begin{cases}\Re\cL(\xi\Delta,n)&\text{if $(n-1)g$ is even},\\\\ i\\,\Im\cL(\xi\Delta,n)&\text{if $(n-1)g$ is odd},\end{cases}$$

with $\cL(\xi\Delta,s)$ the Lerch zeta function associated with $\xi$.

This conjecture predicts that the polylogarithm class specializes to the positive-integer Lerch zeta values, extending the classical three-punctured-projective-line picture. It depends on the conjectural equivariant plectic Hodge theory and is not resolved in the supplied text.