= Solution
The <contracted Bianchi identity> gives $\nabla^aR_{ab}=\tfrac12\nabla_bR$. Combining the proposed equation with <stress-energy conservation> therefore imposes $\nabla_bR=0$. Taking its trace gives $R=\kappa T$, with $T=g^{ab}T_{ab}$, so for nonzero $\kappa$ the matter trace must also be constant. Ordinary matter need not satisfy this extra condition. For example, if asymptotic flatness forces the constant <scalar curvature> to vanish, the proposal would require $T=0$ everywhere.
\b[The problem is an unjustified restriction on the matter trace], rather than the absence of every possible solution. The divergence-free <Einstein tensor> avoids it:
$$
\boxed{G_{ab}=R_{ab}-\frac12Rg_{ab},\qquad\nabla^aG_{ab}=0.}
$$
The <Einstein field equations> $G_{ab}+\Lambda g_{ab}=\kappa T_{ab}$, with constant <cosmological constant> $\Lambda$, are consequently compatible with <stress-energy conservation>.
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