Infinitude conjecture for approximating fiber groups

Let $X_0$ be a nontrivial chain or ring of curves of genus greater than $0$, let $g$ be its arithmetic genus, and let $X_I$ be the family of curves over an interval described in the source. For a sufficiently small subgroup $M^f\subset M_D$, let $S_0$, $S_1$, and $S_k$ be the corresponding sets of vanishing cycles. Infinitude conjecture. There exist a finite-index subgroup $M_j^f\subset Map(g)_f$ and an integer $A>1$ such that

$$\pi_g/\bigl(M_j^f s_i^3,M_j^f s_k^{3^A}\bigr)$$

is infinite for $s_i\subset S_0\cup S_1$ and any finite subset $s_k\in\pi_g$. The statement is part of the proposed construction of groups that could yield counterexamples to Shafarevich's conjecture; its resolution status is not given.