Gross's characteristic-polynomial conjecture for the p-adic regulator matrix

Let $R=\{\mathfrak p_1,\ldots,\mathfrak p_r\}$, let $E_R^*$ be the group of totally positive $R$-units, let $\vartheta\in H_{n+r-1}(E_R^*,\mathbf Z)\cong\mathbf Z$ be a generator, and let $\kappa_{\chi,\lambda}\cap\vartheta\in H_r(E_R^*,\operatorname{Meas}(F_R,K))$ be the cap-product class associated with the Eisenstein cocycle. Let $c_o$ and $c_{to+\ell}$ be the cup-product cocycles defined from $o_{\mathfrak p_i}$ and $t o_{\mathfrak p_i}+\ell_{\mathfrak p_i}$, and let $\mathscr M_p(\chi)$ be Gross's regulator matrix. Gross's characteristic-polynomial conjecture.

$$\det(t\cdot 1_r-\mathscr M_p(\chi))=\frac{c_{to+\ell}\cap(\kappa_{\chi,\lambda}\cap\vartheta)}{c_o\cap(\kappa_{\chi,\lambda}\cap\vartheta)}.$$

This conjecture packages the regulator identities into a single characteristic-polynomial formula. Its denominator is nonzero by the preceding construction, while the asserted equality remains conjectural in the source.