The auxiliary-character $p$-adic Stark conjecture

Let $\eta,\psi\in\mathcal{W}(\mathbb{C}_p)$ be finite-order characters of orders $p^m$ and $p^n$, respectively. Let $M_m$ and $M_n$ be the fixed fields of the kernels of $\rho\otimes\eta$ and $\rho\otimes\psi$, and let $k_m$ and $k_n$ be obtained by adjoining the values of $\chi$, $\alpha$, and $\zeta_{p^{m+1}}$, respectively $\zeta_{p^{n+1}}$, to $\mathbb{Q}$. The $p$-adic Stark conjecture. There exist units $u_{\chi\eta,\alpha}^*\in k_m\otimes\mathscr{O}_{M_m}^\times$ and $u_{\chi\psi,\alpha}^*\in k_n\otimes\mathscr{O}_{M_n}^\times$ such that

$$L_p(\chi,\alpha,\psi\omega,\eta\omega,0)=\frac{(1-\beta\psi(p))\left(1-\frac{\psi^{-1}(p)}{\alpha p}\right)\frac{\tau(\psi^{-1})}{p^{n+1}}}{(1-\beta\eta(p))\left(1-\frac{\eta^{-1}(p)}{\alpha p}\right)\frac{\tau(\eta^{-1})}{p^{m+1}}}\frac{\log_p(u_{\chi\psi,\alpha}^*)}{\log_p(u_{\chi\eta,\alpha}^*)}.$$

Here $\tau(\psi^{-1})$ and $\tau(\eta^{-1})$ are the Gauss sums associated to $\psi^{-1}$ and $\eta^{-1}$. This choice-independent formulation is introduced because the unnormalized one-variable $p$-adic $L$-function depends on auxiliary choices. The source does not provide a resolution status for this conjecture.