The conjectural description of framed diffeomorphisms of the 3-torus

Let $cmathbb T^3$ be the 3-torus, let $cmathrm{Diff}(cmathbb T^3)$ denote its continuous diffeomorphism group, and let $cmathrm{Diff}^{cmathrm{fr}}(cmathbb T^3,cvarphi)$ denote the continuous group of diffeomorphisms preserving a framing $cvarphi$. Write $cmathrm{MCG}(cmathbb T^3)$ and $cmathrm{MCG}^{cmathrm{fr}}(cmathbb T^3,cvarphi)$ for the corresponding mapping class groups.

The 3-torus diffeomorphism conjecture. There are identifications among continuous groups:

$$\operatorname{{\sf Diff}}(\mathbb T^3) \simeq \mathbb T^3 \rtimes \operatorname{{\sf GL}}_3(\mathbb Z)$$

and, for any framing $cvarphi$ of $cmathbb T^3$,

$$\operatorname{{\sf Diff}}^{\operatorname{{\sf fr}}}(\mathbb T^3,cvarphi) \simeq \Bigl( \mathbb T^3 \rtimes \Omega \bigl( \operatorname{{\sf SL}}_3(\mathbb R)_{/\operatorname{{\sf SL}}_3(\mathbb Z)} \bigr) \Bigr) \times \Bigl( \Omega^2 \mathbb S^3 \times \Omega^3 \mathbb S^3 \Bigr)^3 \times \Omega^4 \mathbb S^3.$$

In particular, there is an identification ${\sf MCG}(\mathbb T^3) \cong \operatorname{{\sf GL}}_3(\mathbb Z)$ under which the image of the canonical homomorphism ${\sf MCG}^{\operatorname{{\sf fr}}}(\mathbb T^3,cvarphi) \to {sf MCG}(\mathbb T^3)$ is $\operatorname{{\sf SL}}_3(\mathbb Z)$, with kernel isomorphic to $\mathbb Z^3 \times {\mathbb Z_{/2\mathbb Z}}^2$. This is expected to follow from the methods used for the framed 2-torus, together with Smale's conjecture, proved by Hatcher; the supplied text does not establish the claim.