The Strong Main Conjecture for modular towers

Let $J$ denote the relevant modular-tower data, let $P_{\mathbf C}$ be the exceptional set of primes attached to the conjugacy classes $\mathbf C$, and let $\mathcal V_p(J)$ be the indicated collection of rank-one modules or tower data at $p$. Write $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ for the corresponding inner reduced Hurwitz space.

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 genus $0$ or $1$ components.

This is the strong modular-tower analogue of the behavior of modular curves: low-genus components should occur only finitely often away from the exceptional primes. The source presents it as an expectation and does not state a resolution.