Proposition: Conservation and trace do not imply sourcing

A symmetric, conserved, traceless tensor on a background need not source that background through (eq:einstein).

Proof. In a flat four-dimensional chart with metric $\eta =\mathrm{diag}(-1,1,1,1)$, let

It is constant and symmetric, its covariant divergence vanishes, and ${\eta }^{\mu \nu }{\mathsf{t}}_{\mu \nu }=0$. For $\Lambda =0$, the left side of (eq:einstein) is zero while the right side with $\mathsf{T}=\mathsf{t}$ is nonzero.