  We show that a properly convex projective structure $\mathfrak{p}$ on a
closed oriented surface of negative Euler characteristic arises from a Weyl
connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem
as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$
admits a compatible Weyl connection if and only if a certain holomorphic curve
exists. Turning this non-linear PDE into a transport equation, we obtain our
result by applying methods from geometric inverse problems. In particular, we
use an extension of a remarkable $L^2$-energy identity known as Pestov's
identity to prove a vanishing theorem for the relevant transport equation.
