Conjecture on polynomial-degree covers detecting nonsolvable Torelli images

Let $S$ be a surface and let $H$ be a nonelementary subgroup of its mapping class group. A cover $\widetilde{S}$ to which $H$ lifts is a cover on which every element of $H$ admits a lift. The Torelli homomorphism is the homological representation of the lifted mapping class group, and the wordlengths of generators are measured in a fixed generating set. Polynomial-cover conjecture. There is a cover $\widetilde{S}$ to which $H$ lifts such that the image of $H$ under the Torelli homomorphism is not solvable, and the degree of the cover is at most polynomial in the sum of the wordlengths of the generators of $H$. This is presented as a proposed key tool for proving generic pseudo-Anosov behavior in subgroups of the Torelli group; the source gives no resolution.