The main conjecture for the noncommutative Selmer group of an elliptic curve

Let $E/\mathbb Q$ be an elliptic curve, let $p\geq5$ be a prime at which $E$ has good ordinary reduction, and put $F_{\infty}=\mathbb Q(E_{p^{\infty}})$. Let $G=\operatorname{Gal}(F_{\infty}/\mathbb Q)$ and $H=\operatorname{Gal}(F_{\infty}/\mathbb Q^{\rm cyc})$. Assume that $X(E/F_{\infty})\in\mathfrak M_H(G)$, and grant the p-adic $L$-function conjecture above. A characteristic element of $X(E/F_{\infty})$ is an element of $K_1(\Lambda(G)_{S^*})$ representing its class in the relevant relative $K$-group.

The main conjecture. The p-adic $L$-function ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ is a characteristic element of $X(E/F_{\infty})$.

This is the proposed noncommutative main conjecture, with the existence and interpolation property of ${\cal L}_E$ taken from the preceding conjecture. The paper says that it has deep arithmetic consequences but gives no proof in the non-CM setting.