Proposition: Stress identities do not impose Einstein dynamics

Symmetry, conservation, and zero trace of a tensor on the prescribed host do not imply Einstein's equation.

Proof. In a regular flat chart choose a nonzero constant null covector $\ell$ and $t_{\mu \nu }={\ell }_{\mu }{\ell }_{\nu }$. It is symmetric, conserved, nonzero, and traceless, while the local Einstein tensor vanishes. For nonzero gravitational coupling the proposed Einstein equation fails.