= Solution
Four-dimensional <Hodge duality> identifies $\gamma^{mn}\gamma_5$ with a nonzero convention-dependent multiple of $\epsilon^{mnpq}\gamma_{pq}$. Every $C\gamma_{pq}$ is symmetric by the <four-dimensional Majorana bilinear vanishing criterion>; taking this linear combination preserves symmetry. Hence
$$
\boxed{\bar\lambda\gamma^{mn}\gamma_5\lambda=0.}
$$
The phase in the duality formula depends on the definition of $\gamma_5$ and orientation, but does not affect the vanishing result.
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