Darmon's refined class number formula for real quadratic fields

Suppose $F$ is a real quadratic field and $K_n=F(\boldsymbol{\mu}_n)$, where $n$ is prime to the conductor of $F/\mathbf{Q}$. Let $\theta'_n\in K_n^\times\otimes\mathbf{Z}[\operatorname{Gal}(K_n/F)]$ be Darmon's Stickelberger-type element, and let $R_n$ be the regulator defined from the chosen bases of the relevant unit and augmentation-ideal modules. Writing $\tilde\theta'_n$ for the image of $\theta'_n$ in $F(\boldsymbol{\mu}_n)^\times\otimes I_n^r/I_n^{r+1}$, let $s$ be the number of prime divisors of $n/n_+$, let $r$ be the number of prime factors of $n_+$, and let $h_n$ be the order of $\operatorname{Pic}(\mathcal{O}_F[1/n])$. Darmon's refined class number formula. For every $n\in\mathcal{N}$, one has

$$\tilde\theta'_n=-2^s h_nR_n\quad\text{in }(F(\boldsymbol{\mu}_n)^\times/\{\pm1\})\otimes I_n^r/I_n^{r+1}.$$

This is a leading-term refinement of the class number formula involving first derivatives of $L$-functions. The paper proves most of the conjecture using Kolyvagin systems, reducing the general case to $n=1$ and then to classical formulas; the stated leading-term identity is the conjectural form under discussion.