A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel transport maps are angle preserving with respect to $[g]$.

A torsion-free connection $\nabla$ on $TM$ is called projectively flat if for every point $p \in M$ there exists a neighbourhood $U_p$ and a diffeomorphism $\psi : U_p \to V \subset \mathbb{R}^n$ so that the geodesics of $\nabla$ contained in $U_p$ are mapped onto (segments of) straight lines. It is a classical result that the Levi-Civita connection of a Riemannian metric is projectively flat if and only if the metric has constant sectional curvature.

Is there an example of a non-Riemannian projectivley flat Weyl connection on a closed oriented surface of genus $g\geqslant 2$?