The Harris–Venkatesh conjecture for weight-one newforms

Let $f=\sum_n a_nq^n$ be a newform of weight $1$ and level $\Gamma_1(N)$. Let $\rho_f$ be its Deligne–Serre representation, let ${\mathrm{Ad}}(\rho_f)$ be the trace-free adjoint representation, and let

$$\mathcal{U}({\mathrm{Ad}}(\rho_f))={\mathrm{Hom}}_{{\mathrm{Gal}}(E/\mathbb{Q})}({\mathrm{Ad}}(\rho_f),\mathcal{O}_E^\times\otimes\mathbb{Z}[f])$$

be its associated dual space of units. For each prime $p\nmid 6N$, write ${\mathrm{Reg}}_{(\mathbb{Z}/p\mathbb{Z})^\times}$ for the $(\mathbb{Z}/p\mathbb{Z})^\times$-regulator and $\lVert f\rVert^2_{(\mathbb{Z}/p\mathbb{Z})^\times}$ for the Harris–Venkatesh norm. The Harris–Venkatesh conjecture. There exist $u\in\mathcal{U}({\mathrm{Ad}}(\rho_f))\otimes\mathbb{Q}$ and a prime $p_0$ such that, for all primes $p\geq p_0$,

$$\lVert f\rVert^2_{(\mathbb{Z}/p\mathbb{Z})^\times}={\mathrm{Reg}}_{(\mathbb{Z}/p\mathbb{Z})^\times}(u).$$

Equivalently, the Serre-duality pairing with the derived Hecke operator satisfies

$$\left\langle f^*,T_{\mathbb{Z}_p,N}(f)\right\rangle_{\mathrm{SD}}={\mathrm{Reg}}_{(\mathbb{Z}/p\mathbb{Z})^\times}(u).$$

This conjecture relates the modulo-$p$ derived Hecke action on weight-one forms to Stark-type units and regulators; the supplied text gives no evidence that it has been resolved.