Joint integrability in the exceptional Anosov alternative

Determine whether the second alternative in Theorem 1, namely the existence of a nonzero continuous invariant stable subbundle integrating jointly with the strong unstable distribution, can be further promoted to joint integrability of the full strong stable and strong unstable distributions; likewise, determine whether the $su$-degenerate case of the addendum admits this promotion.

Background

Theorem 1 establishes a dichotomy for transitive Anosov flows with equivalent strong unstable foliations when the model flow has a $C^1$ strong unstable distribution. Either the flows are conjugate up to a constant time rescaling, or there exists a nonzero continuous $DX^t$-invariant subbundle $E\subset E_X^s$ that integrates to an invariant foliation, with $E\oplus E_X^u$ integrating to an invariant foliation subfoliated by that foliation and the strong unstable foliation.

The addendum removes the $C^1$ assumption under local $su$-non-degeneracy, but the author explicitly identifies the unresolved issue of whether the exceptional alternative in the theorem, or the corresponding $su$-degenerate situation in the addendum, forces full joint integrability of $E_X^s$ and $E_X^u$. This question concerns whether the partial integrability produced by the argument can be strengthened to integrability of the entire strong stable and strong unstable distributions.