= Solution
The conserved <Killing energy> of a geodesic tangent $p^a$ is $E=-K_ap^a$, where $K=\partial_t$. Conservation follows from $p^a\nabla_ap^b=0$ and the <Killing equation>. For the <Kerr metric>, $K^2=g_{tt}=-(1-2Mr/\Sigma)$. Outside the outer <Kerr ergoregion>, $K$ is timelike. Its orthogonal complement is positive definite, so a nonzero causal vector cannot have $K\cdot p=0$. Equivalently, if both are future-directed, $-K\cdot p>0$. Hence
$$
\boxed{E=0\ \Longrightarrow\ \text{the causal geodesic cannot enter }r>r_e(\theta),\quad
r_e=M+\sqrt{M^2-a^2\cos^2\theta}.}
$$
This excludes the open exterior of the stationary-limit surface; on the surface itself a zero-energy null tangent can be proportional to the null stationary vector.
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