Strong finiteness conjecture for Kauffman bracket skein modules

Let $M$ be a compact oriented 3-manifold. An essential subsurface is a subsurface $\Sigma_i\subsetpartial M$ as in the conjecture. Let $\Sigma_1,\dots,\Sigma_k$ be a finite collection of such subsurfaces, and let $F_1,\dots,F_k$ be subspaces of the skein module $\mathcal{S}(M)$.

Strong finiteness conjecture. There exists a finite collection $\Sigma_1,\dots,\Sigma_k$ of essential subsurfaces $\Sigma_i\subsetpartial M$ such that, for each $i$, the dimension of $H_1(\Sigma_i,\mathbb{Q})$ is half of that of $H_1(\partial M,\mathbb{Q})$, and $\mathcal{S}(M)$ is a sum of finitely many subspaces $F_1,\dots,F_k$, where $F_i$ is a finitely generated $\mathcal{S}(\Sigma_i,\mathbb{Q}(A))$-module.

This is presented as a more general formulation of the finiteness conjecture for manifolds with boundary. Its general validity is not established in the source; the paper proves the original finiteness conjecture for a large family of Seifert fibered spaces.