Positive Ricci curvature conjecture for compatible metrics on contact 3-manifolds

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the 3-sphere with its standard contact structure. Positive Ricci curvature conjecture. If

$$Ricci(g)>0,$$

then the universal cover of $(M,\xi)$ is contactomorphic to $(\mathbb{S}^3,\xi_{std})$. This would extend the compatible-metric sphere theorem from the $\frac14$-pinched sectional-curvature setting to positive Ricci curvature, relating global Riemannian curvature to the topology of contact structures.