The motivic Lerch element conjecture for totally noncritical characters

Let $F$ be a totally real field, let $\mathfrak g$ be an ideal, put $\Gamma=\operatorname{Gal}(F(\mathfrak g)/F)$, and let $k$ be an integer. Let $\mathbf{Ler}_{\mathfrak g}(k)$ be the Lerch element in $H^{2g-1}_{\mathscr D^I}(\mathscr T_0[\mathfrak g]_{/\mathbb C},\mathbb R(k_I))$, and let $\chi$ be a character of $\Gamma$ totally noncritical for $k$. Under the stated isomorphism with the determinant of the real Deligne cohomology, the element

$$1\otimes d_F^{1/2}\mathbf{Ler}_{\mathfrak g}(k)\in\mathbb C\otimes_{\chi,\mathbb R[\Gamma]}H^{2g-1}_{\mathscr D^I}(\mathscr T_0[\mathfrak g]_{/\mathbb C},\mathbb R(k_I))$$

Motivic Lerch element conjecture. The image of this element belongs to

$$\bigl(\operatorname{det}_{\mathbb Q[\Gamma]}(H^1_{\mathscr M}(\operatorname{Spec}F(\mathfrak g),\mathbb Q(k))_{\mathrm{fin}})\bigr)_\chi.$$

This conjecture reformulates the motivic nature predicted for the Lerch element in each totally noncritical character component. No resolution is given in the supplied text.