The two-sided limit shadowing characterization of Anosov diffeomorphisms

Let $M$ be the compact manifold under consideration, let $f\in\operatorname{Diff}^1(M)$, and let $\mathcal{TLS}$ denote the set of diffeomorphisms with the two-sided limit shadowing property. Anosov diffeomorphisms are diffeomorphisms for which the whole tangent bundle admits a hyperbolic splitting into uniformly contracted and uniformly expanded invariant subbundles. Two-sided limit shadowing conjecture.

$$\mathcal{TLS}\text{ is equal to the set of Anosov diffeomorphisms}.$$

The paper proves that the $C^1$-interior of $\mathcal{TLS}$ is the set of transitive Anosov diffeomorphisms, and notes that this conjecture would imply transitivity of every Anosov diffeomorphism, a longstanding open problem. The conjecture itself is not resolved in the supplied text.