The nonsymmetric skein-module localization conjecture for knots

Let $K\subset S^3$ be a knot. Let $\widehat{K}_q(S^3\setminus K)$ be its nonsymmetric skein module, let $\widehat{K}^{\mathrm{loc}}_q(S^3\setminus K)$ be its localization, and let

$$\eta:\widehat{K}_q(S^3\setminus K)\longrightarrow \widehat{K}^{\mathrm{loc}}_q(S^3\setminus K)$$

be the natural localization map. Let $\mathscr{H}_{q,(t_1,t_2,1,1)}$ act naturally on the localized module. The nonsymmetric skein-module localization conjecture. For every knot $K\subset S^3$, the map $\eta$ is injective, and the natural action of $\mathscr{H}_{q,(t_1,t_2,1,1)}$ on $\widehat{K}^{\mathrm{loc}}_q(S^3\setminus K)$ preserves the subspace $\widehat{K}_q(S^3\setminus K)$, namely the image of $\eta$. The claim generalizes the behavior verified for the unknot in the preceding example; the source gives no resolution status.