The Mazur–Merel version of the Strong Main Conjecture

Let $J$, $P_{\mathbf C}$, $\mathcal V_p(J)$, and $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ be as in the Strong Main Conjecture.

Mazur–Merel version of the Strong Main Conjecture. Assuming the hypotheses of the Strong Main Conjecture, for all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$ give a space $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ having a rational point.

This strengthens the low-genus assertion by requiring finiteness of rational points among the relevant tower spaces. The source says that the weak conjecture and the Strong Main Conjecture imply the corresponding version, but gives no resolution.