The Eisenstein $p$-adic $L$-function conjecture for the adjoint of a weight one modular form

Let $g$ be a weight one modular form, let $g^*$ be its dual, and let $u_1$ denote the unit attached to the associated Galois representation. Let $\kappa(g_{\alpha},g_{1/\alpha}^*)$ be the corresponding cohomology class, and let $L$ be the coefficient field. The Eisenstein $p$-adic $L$-function conjecture. There exists an analytic $p$-adic $L$-function $L_p^{\operatorname{Eis}}(g,g^*,s)$, appropriately related with the cohomology class $\kappa(g_{\alpha},g_{1/\alpha}^*)$ via the theory of Perrin-Riou maps, such that

$$L_p^{\operatorname{Eis}}(g,g^*,1)=\log_p(u_1)\pmod{L^{\times}}.$$

This conjecture seeks a refinement of the preceding result on the arithmetic of the adjoint of a weight one modular form, replacing the simultaneous appearance of the unit and the associated $p$-unit by a $p$-adic $L$-function whose special value encodes information about the unit alone. Its status is not resolved in the supplied source.