Greenberg–Ochiai's specialized main conjecture at a height-one prime

Let $\mathbb{I}$ be a UFD and Gorenstein, assume $\chi|_{((\mathbb{Z}/p\mathbb{Z})^{\times})}\neq\mathbf{1},\omega$, $p\nmid\varphi(N)$, and that the Eisenstein ideal $\mathbb{J}$ is principal. Let $\mathbb{T}$ be a stable free lattice, let $\mathfrak{p}_i$ be a height-one prime, and put $\mathcal{A}[\mathfrak{p}_i]$ for the corresponding specialization. Greenberg–Ochiai's height-one specialization conjecture. If $(\operatorname{Sel}_{\mathcal{A}[\mathfrak{p}_i]})^{\lor}$ is a torsion $\mathcal{R}/\mathfrak{p}_i$-module, then

$$\operatorname{char}_{\mathcal{R}/\mathfrak{p}_i}(H^0(\mathbb{Q},\mathcal{A}[\mathfrak{p}_i])^{\lor})^{-1}\operatorname{char}_{\mathcal{R}/\mathfrak{p}_i}(\operatorname{Sel}_{\mathcal{A}[\mathfrak{p}_i]})^{\lor}=\left(L_p(\mathcal{T}/\mathfrak{p}_i\mathcal{T})\right).$$

This is the main-conjecture formula after specialization at a height-one prime. The supplied text gives no resolution.