Proposition: Conservation and trace are not Einstein dynamics

On a prescribed metric, symmetry, conservation, and zero trace of a tensor do not imply Einstein's equation.

Proof. In a regular flat chart choose a nonzero constant null covector $k$ and set $t_{\mu \nu }=k_{\mu }{k}_{\nu }$. It is symmetric, constant and conserved, nonzero, and traceless. The chart's Einstein tensor is zero. Thus $G_{\mu \nu }=8\pi G_{\mathrm{N}}{t}_{\mu \nu }$ fails when $G_{\mathrm{N}}\ne 0$. No statement on a distributional stress tensor at the defect is needed.