Proposition: No Einstein implication

Symmetry, conservation, and zero trace of a tensor on the prescribed metric do not imply Einstein dynamics.

Proof. In a regular flat chart take a nonzero constant null covector $k$, and set $t_{\mu \nu }=k_{\mu }{k}_{\nu }$. It is symmetric, conserved, nonzero, and traceless. The chart Einstein tensor is zero, so $G_{\mu \nu }=8\pi G_{\mathrm{N}}{t}_{\mu \nu }$ fails for $G_{\mathrm{N}}\ne 0$.