There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor

We show that if two 4-dimensional metrics of arbitrary signature on one manifold are geodesically equivalent (i.e., have the same geodesics considered as unparameterized curves) and are solutions of the Einstein field equation with the same stress-energy tensor, then they are affinely equivalent or flat. Under the additional assumption that the metrics are complete or the manifold is closed, the result survives in all dimensions >2.