Harris--Venkatesh conjecture for derived Hecke operators on weight-1 forms

Let $f$ be a Hecke new cusp form of weight $1$ and level $N$, with adjoint Artin representation $\operatorname{Ad}(\rho)$. Let $\mathcal{U}(\operatorname{Ad}(\rho))$ be its dual unit group, let $\mathfrak{S}_p$ denote the Shimura class, let $\operatorname{Tr}_p^{Np}$ be the indicated trace operator, and let $f^*$ be the dual newform. For primes $p,\ell\geq 5$ coprime to $N$, write $\log_\ell:\mathbb{F}_p^\times\twoheadrightarrow\mathbb{Z}/\ell^t\mathbb{Z}$ for a fixed discrete logarithm, where $\ell^t$ is the exact power of $\ell$ dividing $p-1$, and let $\operatorname{Reg}_{\mathbb{F}_p^\times}$ be the finite-field regulator map.

Harris--Venkatesh conjecture. There are an element $u\in\mathcal{U}(\operatorname{Ad}(\rho))$ and a positive integer $m$ such that, for any primes $p,\ell\geq 5$ coprime to $N$,

$$m\cdot\log_\ell\mathfrak{S}_p\left(\operatorname{Tr}_p^{Np}\left(f(z)f^*(pz)\right)\right)=\log_\ell\operatorname{Reg}_{\mathbb{F}_p^\times}(u).$$

This conjecture predicts that the Shimura-class expression obtained from a weight-1 newform agrees, after a fixed integral scaling and discrete logarithm, with the reduction of a single Stark unit. The supplied text presents it as the Harris--Venkatesh conjecture away from the primes $2$ and $3$, but gives no resolution.