The $p$-adic Stark conjecture in the cyclotomic tower

For each $n\ge 0$, let $\mathbb{Q}_n$ be the $n$th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, let $M_n=M\mathbb{Q}_n$, let $\Delta_n=\operatorname{Gal}(M_n/\mathbb{Q})$, and let $U_n$ be the unit group specified in the source. Let $k_n$ be obtained from $k$ by adjoining the $p^{n+1}$st roots of unity. For characters $\eta,\psi$ of the cyclotomic Galois groups, let $(\rho\eta)^*$ denote the indicated isotypic representation, and let $A^{(\rho,\eta)^*}$ denote its isotypic component; write $A^{\delta_p=\alpha}$ for the $\alpha$-eigenspace of geometric Frobenius at $p$. The cyclotomic $p$-adic Stark conjecture. If $\psi,\eta\in\mathcal{W}(\mathbb{C}_p)$ have orders $p^n$ and $p^m$, respectively, with $m,n\ge 1$, then there exist units

$$u_{\chi\psi,\alpha}^*\in(k_n\otimes U_n)^{(\rho\psi)^*,\delta_p=\alpha},\qquad u_{\chi\eta,\alpha}^*\in(k_m\otimes U_m)^{(\rho\eta)^*,\delta_p=\alpha}$$

for which

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

This is the more precise unit-theoretic formulation in the cyclotomic tower, incorporating the Frobenius eigenvalue $\alpha$. The source gives no evidence that it has been proved or refuted.