The Eisenstein-cocycle formula for Gross's p-adic regulator

Let $R$ be the set of primes of $F$ above $p$ that split in $H$, let $J\subset R$, let $E_R^*$ be the group of totally positive $R$-units, let $F_R=\prod_{\mathfrak p\in R}F_{\mathfrak p}$, and let $K$ contain the values of $\chi$. Let $\kappa_\chi\in H^{n-1}(E_R^*,\operatorname{Meas}(F_R,K))$, let $c_{\ell,J},c_o\in H^r(E_R^*,C_c(F_R,K))$ be the cocycles defined using the homomorphisms $\ell_{\mathfrak p}$ and $o_{\mathfrak p}$, and choose a generator $\vartheta\in H_{n+r-1}(E_R^*,\mathbf Z)\cong\mathbf Z$. Define

$$\mathscr{R}_p(\chi)_{J,\mathrm{an}}:=(-1)^{\#J}\frac{c_{\ell,J}\cap(\kappa_\chi\cap\vartheta)}{c_o\cap(\kappa_\chi\cap\vartheta)}\in K.$$

The Eisenstein-cocycle regulator conjecture. For every subset $J\subset R$,

$$\mathscr{R}_p(\chi)_J=\mathscr{R}_p(\chi)_{J,\mathrm{an}}.$$

The denominator is nonzero because it equals $L(\chi,0)\prod_{\mathfrak p\in R_0}(1-\chi(\mathfrak p))$ up to sign. The conjecture gives an analytic Eisenstein-cocycle expression for the regulators entering Gross's leading-term conjecture.