Derived reciprocity conjecture for the Kato class in the $\alpha_g\beta_h=1$ case

Assume $\alpha_g\beta_h=1$, and let $P,Q$ be the points in $(E(H)\otimes V_{gh}^{\vee})^{G_{\mathbb Q}}$. Let $\kappa'(f,g_{\alpha},h_{1/\beta})$ be the derived cohomology class, let $\mathcal L$ be the relevant $\mathcal L$-invariant, and let $\Xi_{g_{\alpha}}$, $\Omega_{h_{1/\beta}}$, and $u_{g_{\alpha}}$ be the associated periods and Gross--Stark unit. Derived reciprocity conjecture. In $H_{\operatorname{f}}^1(\mathbb Q,V_{fgh})$,

$$\kappa'(f,g_{\alpha},h_{1/\beta})=\frac{\mathcal L}{\Xi_{g_{\alpha}}\cdot\Omega_{h_{1/\beta}}}\cdot\frac{\log_p(P_{\beta\beta})\cdot Q-\log_p(Q_{\beta\beta})\cdot P}{\log_p(u_{g_{\alpha}})}\pmod{L^{\times}}.$$

This is the conjectural analogue of a derived reciprocity theorem: the ordinary reciprocity law degenerates in this case, and the displayed formula is expected to express the derived class in terms of the points. Its general validity remains open.