The reciprocity conjecture for Kolyvagin determinants

Let $T$ be the $p$-adic Galois representation and let $\mathfrak{K}(T)$ be the $\Lambda$-module of Kolyvagin determinants. For an even Dirichlet character $\theta$ and $\underline{I}\in\mathfrak{I}_p$, let $\mathfrak{M}_\theta^{\underline{I}}$ be the regulator matrix defined from Kolyvagin determinant classes. The reciprocity conjecture. There exists a unique non-zero element $\mathfrak{c}=\mathfrak{c}_1\wedge\cdots\wedge\mathfrak{c}_{g_-}\in\mathfrak{K}(T)$ such that

$$\det\left(\mathfrak{M}_\theta^{\underline{I}}(\mathfrak{c})\right)=L_{\{p\}}(\mathcal{M}^*(1),\theta^{-1},1)\frac{\Omega_{\mathcal{M}(\theta)^*(1),p}(\underline{I})}{\Omega_{\mathcal{M}(\theta)^*(1)}(\underline{I})}$$

for all $\underline{I}$ and $\theta$ as specified. This conjecture supplies the special Kolyvagin determinant needed for the paper's one-sided results toward the signed and Perrin-Riou main conjectures; its status is not resolved in the supplied text.