Injectivity conjecture for the geodesic Wilson loop operator

Let $(M,g)$ be a closed odd-dimensional Riemannian manifold with Anosov geodesic flow. Let $\boldsymbol{A}$ denote the moduli space of the relevant connections, let $\boldsymbol{ obreakmathcal{C}}^{\sharp}$ denote the set of primitive closed geodesics, and define the geodesic Wilson loop operator by

$$\mathbf{W}: \mathbf{A} \to \ell^\infty(\mathcal{C}^{\sharp}), \qquad \mathbf{W}(a)(c):= \operatorname{Tr}(\operatorname{Hol}_{\nabla^{E}}(c)).$$

Wilson-loop injectivity conjecture. If $(M,g)$ is a closed odd-dimensional Riemannian manifold with Anosov geodesic flow, then the Wilson loop operator

$$\mathbf{W}: \mathbf{A} \to \ell^\infty(\mathcal{C}^{\sharp})$$

is injective.

This conjecture reduces the associated inverse problem for connection Laplacians to recovering a connection from traces of holonomies along primitive closed geodesics. The supplied source does not state that the conjecture has been resolved.