Conjectures on the comparison of geometric deformation modules

Let $M$ be a $3$-manifold, let $\mathcal{W}_r(M)$ and $\mathcal{W}_r^+(M)$ denote the degree-$r$ geometric deformation modules and their integrability quotients, and let $\mathcal{W}(M)$ and $\mathcal{W}^+(M)$ be their graded versions. A standard link is denoted $K_{\alpha}$ for $\alpha\in\mathfrak{b}(M)$, and an extended standard link is a standard link together with iterated Hopf links, subject to the stated symmetries.

Comparison conjectures. (a) The projection

$$\mathcal{W}_r(S^3)\rightarrow\mathcal{W}_r^+(S^3)$$

is an isomorphism for $r=3$ but is not an isomorphism for $r\geq 4$. (b) $M$ is simply connected if and only if

$$\mathcal{W}_3(M)\rightarrow\mathcal{W}_3^+(M)$$

is an isomorphism. (c) If $M$ is a submanifold of a rational homology sphere, a minimal generating set for $\mathcal{W}(M)$ is defined by all extended standard links. (d) If $M\neq S^3$ is a Seifert-fibred $3$-manifold, then $\mathcal{W}(M)$ has torsion.

The surrounding discussion motivates these assertions through possible differences between the geometric deformation modules and their integrability quotients, and through torsion in Seifert-fibred manifolds. The source does not provide resolution evidence for the grouped claims.