Darmon–Harris–Venkatesh Stark-unit conjecture for the derived Hecke pairing

Let $R$ be the ring of integers of $L$, with $6N$ inverted, and let $g$ be the weight-one form with associated representation $\rho_g$. Define

$$G(z):=\mathsf{Tr}^{Nd}_{N}(g(z)g^*(Nz))\in S_2(N;R)=H^0(X_0(N)_{/R},\Omega^1).$$

Let $\mathfrak{S}$ be the Shimura class, let $\langle G,\mathfrak{S}\rangle\in R/p^t$ be the pairing obtained from Serre duality, let $H$ be the finite extension of $\mathbb{Q}$ cut out by $\operatorname{Ad}(\rho_g)$, and set

$$U_g:=({\mathcal O}_H^\times\otimes\operatorname{Ad}^*(\rho_g)^\circ)^{G_{\mathbb Q}}.$$

For a prime $\mathcal{N}$ of $H$ above $N$, let

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

be the reduction map defined using the Frobenius element $\sigma_N$. Darmon–Harris–Venkatesh Stark-unit conjecture. There exists an integer $m=m_g\geq1$ and $u_g\in U_g$ such that, for all primes $N$ and $p$ as above,

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

This conjecture relates a derived-Hecke or Serre-duality pairing to the discrete logarithm of a Stark unit associated with $g$. The source presents it as a version of the main conjecture of the cited work and does not state that it has been resolved.