Equivalence of the topological Anosov definition with classical hyperbolicity criteria

Let $M$ be a manifold and let $\theta$ be a continuous flow on $M$. A standard topological-hyperbolicity criterion consists of a continuous invariant splitting

$$TM=E^s_\theta\oplus E^u_\theta\oplus E^c_\theta,$$

where $E^c_\theta$ is the one-dimensional bundle tangent to the flow orbits, together with a continuous Riemannian metric for which $E^s_\theta$ is uniformly contracted exponentially by the forward flow $\theta_t$ for $t>0$, and $E^u_\theta$ is uniformly contracted exponentially by the backward flow $\theta_{-t}$ for $t>0$.

Equivalence with classical definitions. A continuous flow $\theta$ is a topological Anosov flow according to the source's definition if and only if it satisfies such standard criteria for topological hyperbolicity.

This conjecture asks whether the approximation-based definition agrees with classical intrinsic hyperbolicity. The source presents it as a fundamental connection question and gives no resolution.