Conjecture on geometric deformations and the Hoste–Przytycki module

Let $M$ be an oriented $3$-manifold. Write $\mathcal{W}_{\centerdot}(M;\mathbb{Z})$ for the universal geometric deformation module, $\mathcal{C}_{\centerdot}(M)$ for the Hoste–Przytycki skein module, and let $\gamma$ denote the universal pairing. A geometric $R$-deformation is a deformation obtained over a ring $R$.

Geometric deformation conjecture. (a) For each oriented $3$-manifold, the epimorphism

$$\mu_{\mathcal{C}_{\centerdot}(M)}:\mathcal{W}_{\centerdot}(M;\mathbb{Z})\rightarrow\mathcal{C}_{\centerdot}(M)$$

is not an isomorphism. (b) If each essential torus map into $M$ is homotopic into $\partial M$, then $\mathcal{W}(M;\mathbb{Z})$ is free and the universal pairing is trivial. (c) For a given $3$-manifold $M$, if $\gamma$ is trivial, then each geometric $R$-deformation of $\gamma$ is trivial.

These claims concern the structure of the universal geometric deformation and its relationship with the Hoste–Przytycki module. The source presents them as conjectural expectations; their general status is not established there.