Harris--Venkatesh plus Stark conjecture for weight-1 newforms

Let $f$ be a newform of weight $1$ and level $N$, with associated representation $\rho_f$. Let $\mathcal{U}(\operatorname{Ad}(\rho_f))$ be the equivariant unit module, and let $\|f\|_{\mathbb{R}}^2$ and $\|f\|_{\mathbb{F}_p^\times}^2$ denote respectively the Petersson norm and the Harris--Venkatesh norm defined using the Shimura class. Let

$$\operatorname{Reg}_{\mathbb{R}}:\mathcal{U}(\operatorname{Ad}(\rho_f))\otimes\mathbb{Q}\longrightarrow\mathbb{R}\otimes\mathbb{Q}[\chi_{\operatorname{Ad}(\rho_f)}]$$

and

$$\operatorname{Reg}_{\mathbb{F}_p^\times}:\mathcal{U}(\operatorname{Ad}(\rho_f))\otimes\mathbb{Z}_{(p-1)}\longrightarrow\mathbb{F}_p^\times\otimes\mathbb{Z}\left[\chi_{\operatorname{Ad}(\rho_f)},\frac1{6N}\right]$$

be the corresponding regulators. Harris--Venkatesh plus Stark conjecture. There is a unique element $u_f\in\mathcal{U}(\operatorname{Ad}(\rho_f))\otimes\mathbb{Q}$ such that

$$\|f\|_{\mathbb{R}}^2=\operatorname{Reg}_{\mathbb{R}}(u_f),$$

compatibly with conjugations of $f$ under $\operatorname{Aut}(\mathbb{C})$, and, for each sufficiently large prime $p$,

$$\|f\|_{\mathbb{F}_p^\times}^2=\operatorname{Reg}_{\mathbb{F}_p^\times}(u_f).$$

This is the unified conjecture claimed in the paper; the paper proves it for imaginary dihedral forms, while the general weight-$1$ case remains open.