Shintani's exact analytic formula for Brumer–Stark units

Let $F$ be a totally real field, let $\mathfrak{n}$ be a nonzero ideal, let $F(\mathfrak{n})$ be its narrow ray class field, and let $H$ be the maximal CM subfield in which $\mathfrak{p}$ splits completely. Let $\pi$ satisfy $\mathfrak{p}^f=(\pi)$ as in the setup, let $\mathscr{D}$ be a Shintani domain, let $\mathbf{O}=\mathcal{O}_{\mathfrak{p}}-\pi\mathcal{O}_{\mathfrak{p}}$, and let $\nu(\mathfrak{b},\mathscr{D})$ be the associated measure. For the Frobenius element $\sigma_{\mathfrak{b}}\in\operatorname{Gal}(H/F)$, Shintani's exact formula. The associated conjugate of the Brumer–Stark unit satisfies

$$\sigma_{\mathfrak{b}}(u_{\mathfrak{p}})=\epsilon(\mathfrak{b},\mathcal{D},\pi)\,\pi^{\zeta_{S,T}(F(\mathfrak{n})/F,\mathfrak{b},0)}\int_{\mathbf{O}}x\,d\nu(\mathfrak{b},\mathscr{D},x)\in F_{\mathfrak{p}}^*.$$

This is the conjectural exact $p$-adic analytic formula proposed in the cited work; the source explains that a theorem relating the $p$-part of Gross's conjecture implies it up to a root of unity.