GJS conjecture on skein-module dimension and Abouzaid–Manolescu homology

Let $M$ be a closed $3$-manifold. The skein module $S(M)$ is a $[?]$-vector space, and $HP^0_\#(M)$ denotes the degree-zero part of Abouzaid–Manolescu homology $HP_\#(M)$.

GJS conjecture.

$$\dim_{\mathbb{Q}(A)} S(M)=\dim_{\mathbb{C}} HP^0_\#(M).$$

The preceding finiteness theorem shows that the left-hand side is finite, but its computation and its interpretation remain difficult and open. The conjecture proposes that this dimension is governed by Abouzaid–Manolescu homology.