= Solution
After using the <Clifford algebra>, integrations by parts and the <Dirac equation>, every allowed Lorentz-covariant symmetric rank-two expression with one derivative reduces, modulo terms vanishing <on shell> and identically conserved improvements, to an overall multiple of
$$
\bar\psi\gamma^{(\mu}\overleftrightarrow{\partial}^{\nu)}\psi.
$$
Thus the general nontrivial tensor in the stated class is
$$
\widehat T^{\mu\nu}
=C\left[
\bar\psi\gamma^\mu\partial^\nu\psi
+\bar\psi\gamma^\nu\partial^\mu\psi
-(\partial^\nu\bar\psi)\gamma^\mu\psi
-(\partial^\mu\bar\psi)\gamma^\nu\psi
\right].
$$
It is manifestly symmetric. Differentiating it, commuting partial derivatives and using
$$
i\gamma^\rho\partial_\rho\psi=m\psi,
\qquad
i(\partial_\rho\bar\psi)\gamma^\rho=-m\bar\psi,
$$
makes the terms cancel pairwise, proving $\partial_\mu\widehat T^{\mu\nu}=0$ on shell. The normalization $C$ remains free at this stage.
Solved by gpt-5.6-sol high.
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