Harris–Venkatesh conjecture for triple product L-functions

Let $N$ and $p\geq 5$ be primes with $p\mid N-1$. Let $g\in S_1(\Gamma_1(d))$ be a weight-one cuspidal newform of conductor $d$ with $\gcd(Np,d)=1$, let $g^*$ be its dual, and let $G$ be the trace of $g(z)g^*(Nz)$ from $S_2(\Gamma_0(Nd))$ to $S_2(\Gamma_0(N))$. Let $U_g$ be the $R$-module of units cut out by the adjoint of the Artin representation attached to $g$, where $R=\mathcal{O}_g[1/6]$, and let

$$\operatorname{red}_N:U_g\longrightarrow (\operatorname{\mathbf{Z}}/N\operatorname{\mathbf{Z}})^\times\otimes_{\operatorname{\mathbf{Z}}}R$$

be its reduction modulo $N$ map. Fix a surjective homomorphism $\log:(\operatorname{\mathbf{Z}}/N\operatorname{\mathbf{Z}})^\times\to\operatorname{\mathbf{Z}}/p\operatorname{\mathbf{Z}}$, and let $\mathfrak{S}$ be the associated Shimura class. Harris–Venkatesh conjecture. There exists an integer $m_g\geq 1$ and an element $u_g\in U_g$ such that, for all primes $N$ and $p$ as above, in $R/pR$ one has

$$m_g\cdot\langle G,\mathfrak{S}\rangle=\log(\operatorname{red}_N(u_g)).$$

The conjecture relates the Shimura-class pairing with the reduction of a unit arising from the adjoint representation of the weight-one form, and is intended to explain the arithmetic of the associated triple product $L$-functions. Its resolution is not established by the supplied source context.