A prescribed covariant stress-energy tensor need not be conserved

Force density of $F{\eta }^{-1}$. Since the components ${\eta }^{ab}$ are constant in the inertial chart, differentiating componentwise by [F4] gives $f^b={\mathrm{\partial }}_a(F{\eta }^{ab})={\eta }^{ab}{\mathrm{\partial }}_{a}F={\mathrm{\partial }}^{b}F$. Because the matrix ${\eta }^{ab}$ is invertible (it is the inverse of the nondegenerate form $\eta$), $f=0$ on $\mathrm{\Omega }$ if and only if ${\mathrm{\partial }}_{a}F=0$ for every $a$.

Conservation forces constancy. By [F2] the $C^1$ function $F$ is totally differentiable, with derivative the covector of its partial derivatives; if $f=0$, step 1.1 gives $DF=0$, and [F3] gives that $F$ is constant on the connected open set $\mathrm{\Omega }$; conversely a constant $F$ has $f=0$ by step 1.1. Hence $T=F{\eta }^{-1}$ is conserved exactly for constant $F$, and for nonconstant $F$ the continuous force density $f$, with the components computed in step 1.1, is not identically zero although it may vanish at points.

Explicit witness. On $\mathrm{\Omega }={\mathbb{R}}^4$ (identified with an inertial chart) take $F(x)=(x^1{)}^2$, which is smooth and nonconstant. Then ${\mathrm{\partial }}_{0}F={\mathrm{\partial }}_{2}F={\mathrm{\partial }}_{3}F=0$ and ${\mathrm{\partial }}_{1}F=2x^1$, so $f=(0,2x^1,0,0)$: it vanishes exactly on the hyperplane $x^1=0$ and is nonzero at every point off that hyperplane. Thus the prescribed field $T^{ab}=F{\eta }^{ab}$ has nonzero force density on a nonempty open subset of $\mathrm{\Omega }$, so it is not locally conserved, although $f$ vanishes at some points.

Covariance does not imply conservation. Under an inertial transition $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+a$ with $\mathrm{\Lambda }$ a proper orthochronous Lorentz matrix (Inertial frames and inertial charts of the affine Minkowski model), the components of the same tensor transform by the rank-two law $T^{\mathrm{\prime }ab}={\mathrm{\Lambda }}^a{}_c{\mathrm{\Lambda }}^b{}_d{T}^{cd}$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii)); writing $F^{\mathrm{\prime }}(x^{\mathrm{\prime }}):=F({\mathrm{\Lambda }}^{-1}(x^{\mathrm{\prime }}-a))$ for the scalar, one gets $T^{\mathrm{\prime }ab}=F^{\mathrm{\prime }}{\mathrm{\Lambda }}^a{}_c{\mathrm{\Lambda }}^b{}_d{\eta }^{cd}=F^{\mathrm{\prime }}{\eta }^{ab}$, because $\mathrm{\Lambda }{\eta }^{-1}{\mathrm{\Lambda }}^{\mathsf{T}}={\eta }^{-1}$ for a Lorentz matrix. So the expression $F{\eta }^{-1}$ is manifestly covariant; yet by step 2.1 its local conservation is equivalent to constancy of the scalar $F$, which covariance does not require.

The field of step 2.2 is a $C^1$ symmetric type-$(2,0)$ field of the form $F{\eta }^{-1}$ on the connected open set ${\mathbb{R}}^4$ that is not locally conserved, refuting the statement. The example shows that covariance of a prescribed tensor expression and conservation of that tensor are independent properties: conservation is an additional premise. The prescribed field is not claimed to be the stress-energy tensor of any matter model of this page and no choice principle is used.