The orbit-containment criterion for congruence subgroups of mapping class groups

Let $\Gamma$ be a finite-index subgroup of $\operatorname{Mcg}(\Sigma)$, and let $\alpha$ be a simple closed curve on $\Sigma$. A subgroup of $\operatorname{Mcg}(\Sigma)$ is congruence if it contains the kernel of the action on a finite quotient of $\pi_1(\Sigma)$. The orbit-containment conjecture. There exists a congruence subgroup $\Delta\leq\operatorname{Mcg}(\Sigma)$ such that

$$\Delta.\alpha\subseteq\Gamma.\alpha.$$

This condition would, together with the theorem cited in the paper, imply the congruence subgroup property for mapping class groups. It is therefore proposed as a sufficient criterion rather than as the full congruence subgroup conjecture.