Fourth skein module generation conjecture for tangles

Let $M$ be an oriented $3$-manifold, let $R$ be a commutative ring with identity, and let $\mathcal{S}_4(M;R,a,b_0,b_1,b_2,b_3)$ be the fourth skein module defined using the framing relation and the fourth skein relation. For a disk with $2n$ boundary points, regard an $n$-tangle as a relative framed link.

Fourth skein module generation conjecture. The fourth skein module $\mathcal{S}_4(S^3)$ is generated by trivial links; the fourth skein module of $2$-tangles in a disk is generated by tangles with at most one crossing; the fourth skein module of $3$-tangles in a disk is generated by the $40$ basic $3$-tangles described in Figure $3$, with possibly trivial components; and there is a function $h(n)$ such that the fourth skein module of $n$-tangles in a disk is generated by tangles with at most $h(n)$ crossings.

This conjecture generalizes the Montesinos–Nakanishi conjecture, and the source notes that it reduces to that conjecture when $b_1=b_2=0$ and $b_0=-b_3$. The paper later proves the $2$- and $3$-algebraic cases, while the general generation assertion remains conjectural in the supplied text.