Characterization of smoothness for topological Anosov flows

Let $M$ be the underlying manifold, let $\mathcal{TA}\mathcal{F}(M)$ be the class of topological Anosov flows, and let $\operatorname{An}(M)$ denote the smooth Anosov vector fields on $M$. Suppose $\theta\in\mathcal{TA}\mathcal{F}(M)$ has $C^0$-generator $X\in C^0(TM)$, obtained as the $C^0$ limit of approximating generators $X_i$. Let $d_{\mathrm{An},C^1}$ be the stated metric on generators.

Characterization of smoothness. The flow $\theta$ is topologically conjugate to a smooth Anosov flow generated by some $Y\in\operatorname{An}(M)$ if and only if $X$ belongs to the closure of $\operatorname{An}(M)$ in the $d_{\mathrm{An},C^1}$-metric topology, or in a similarly strong topology related to smoothness.

This conjecture seeks an intrinsic criterion distinguishing those topological Anosov flows that arise, up to topological conjugacy, from smooth Anosov flows. The source gives no resolution and leaves the precise strong topology partly unspecified.