= Solution
Decompose the <electric field> into its orthogonal parts $\mathbf E=\mathbf E_\perp+\mathbf E_\parallel$, where $\mathbf E_\parallel=n(n\cdot\mathbf E)$. The <uniaxial conductivity tensor> acts on them as
$$
\mathbf J=\alpha\mathbf E_\perp+(\alpha+\gamma)\mathbf E_\parallel.
$$
If $\mathbf J=0$, its perpendicular and parallel components each vanish. Since $\alpha\ne0$ and $\alpha+\gamma\ne0$, this forces both field components to be zero. The contrapositive gives
$$
\boxed{\mathbf E\ne0\quad\Longrightarrow\quad\mathbf J\ne0}.
$$
Equivalently, the tensor has nonzero <eigenvalues> $\alpha,\alpha,\alpha+\gamma$ and determinant $\alpha^2(\alpha+\gamma)\ne0$.
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