Lemma: Split type implies the cone condition

A split-type distribution satisfies $\mathrm{WF}(u)\subset {\Xi }_n$.

Proof. If all $p_j$ are future causal, their time components are nonnegative. Their sum can vanish only if every time component vanishes, which forces every causal $p_j$ to vanish. The zero total covector is not in a wavefront set. The past case is identical.