Integral equivariant Iwasawa main conjecture for arbitrary sets of primes

Let $F$) be a totally real field, let $p$ be a prime, let $H_{\infty}/F$ be the relevant infinite extension with Galois group $\mathcal{G}$, and let $S$, $S_{ram}$, and $S_p$ denote respectively a set of primes, the primes ramified in the extension, and the primes above $p$. Let $X_S^{-}$ be the minus part of the corresponding Iwasawa module, and let $\Theta_{((S\cap S_{ram})\cup S_p)}^{\{v_0\}}(H_{\infty})$, $Q_v$, $\Tilde{J}_v^{\infty}$, and $R_v$ be the associated equivariant elements or ideals. Here $v_0$ is a prime with $p\nmid v_0$. Integral equivariant Iwasawa main conjecture. For any set $S$ of primes,

$$\operatorname{Fitt}_{\mathbb{Z}_p[[\mathcal{G}]]^-}(X_S^{-})=\left( \Theta_{((S\cap S_{ram})\cup S_p)}^{\{v_0\}}(H_{\infty})\prod_{v\in S_{ram}\setminus (S\cup \{v_0\}\cup S_p)} Q_v \prod_{S_p\setminus S}\Tilde{J}_v^{\infty}\prod_{S\cap S_{ram}}R_v;p\nmid v_0 \right).$$

The conjecture extends the integral equivariant refinement of the Iwasawa main conjecture to the case without an auxiliary nonempty set $T$. The paper notes that the inclusion in one direction follows from the surjection $X_S^{\{v_0\},-}\to X_S^-$; the reverse inclusion remains conjectural.