Rigidity of $\mathit{\text{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on ${\mathbb{T}}^3$

We show that within a $C^1$-neighbourhood $\mathcal{U}$ of the set of volume preserving Anosov diffeomorphisms on the three-torus ${\mathbb{T}}^3$ which are strongly partially hyperbolic with expanding center, any $f\in \mathcal{U}\cap {\mathrm{Diff}}^2({\mathbb{T}}^3)$ satisfies the dichotomy: either the strong stable and unstable bundles $E^s$ and $E^u$ of $f$ are jointly integrable, or any fully supported $u$-Gibbs measure of $f$ is SRB.