Handlebody basis conjecture for the three-variable skein module

Let $\langle G\rangle$ denote the set of conjugacy classes of elements of $G$ different from $1$. For a handlebody $H_n$, let $\mathrm{X}_i\langle\pi_1(H_n)\rangle$ denote the set of sequences of length $i$ with entries in $\langle\pi_1(H_n)\rangle$, and let $\sim$ identify sequences that differ only by permutation. The three-variable skein module is ${\mathcal S}_3(H_n)$, with coefficient ring $Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}]$.

Handlebody basis conjecture. For every handlebody $H_n$, ${\mathcal S}_3(H_n)$ is the free $Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}]$-module with basis

$$\left(\bigcup_{i=1}^{\infty}\mathrm{X}_i\langle\pi_1(H_n)\rangle\right)/\sim.$$

Equivalently, it is the module of polynomials

$$Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}][\langle\pi_1(H_n)\rangle]$$

whose variables come from $\langle\pi_1(H_n)\rangle$, with monomials corresponding to layered families of links in $H_n$. The empty sequence, or polynomial $1$, corresponds to a trivial knot, and a sequence $\gamma_1,\ldots,\gamma_s$ corresponds to a layered family representing $\gamma_1\cdots\gamma_s$.

This conjecture generalizes the stated results for $S^3$ and the solid torus. The source also notes that different sequence representatives can lead to different bases, and that freeness does not follow merely from $\pi_1(M)$ being free, as shown by $S^1\times S^2$.