The Dehn-filling conjecture for mapping class groups

Let $S$ be any surface of finite type, and let $g_1,\dots,g_l\in MCG(S)$. For a subset $A$ of a group, write $\langle\langle A\rangle\rangle$ for its normal closure. Dehn-filling conjecture. There exists $N\in\mathbb{N}-\{0\}$ such that

$$MCG(S)/\langle\langle \{g_i^{N}\}\rangle\rangle$$

is hierarchically hyperbolic. This conjecture seeks a general theory of Dehn-filling-like quotients of hierarchically hyperbolic groups, extending known constructions for mapping class groups and potentially producing hyperbolic quotients and applications to algebraic properties such as residual finiteness and the congruence subgroup property.