Solenoidal injectivity conjecture for the X-ray transform on manifolds of Anosov type

Let $(M,g)$ be a smooth Riemannian manifold of Anosov type. For symmetric $m$-tensors, let $I_m^g$ be the X-ray transform, and call it solenoidal injective when

$$\ker I_m^g=D_g\left(C^{k+1,\alpha}_0(M,\otimes_S^{m-1}T^*M)\right).$$

Here $D_g$ is the symmetrized covariant derivative. Solenoidal injectivity conjecture. The transform $I_m^g$ is solenoidal injective. Solenoidal injectivity is known for non-positively curved manifolds of Anosov type, all surfaces of Anosov type, and real-analytic manifolds in the stated tensorial case. It remains open for general manifolds of Anosov type.