  Let $f$ be a conservative partially hyperbolic diffeomorphism, which is
homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the
stable and unstable bundles of $f$ are jointly integrable if and only if every
periodic point of $f$ admits the same center Lyapunov exponent with $A$. In
particular, $f$ is Anosov. Thus every conservative partially hyperbolic
diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$,
is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on
$\mathbb{T}^3$.
