Proposition

If an antichain $\Sigma$ contains $e$, every directed path starting at $e$ meets $\Sigma$ only at $e$. Thus requiring such a path to meet $\Sigma$ within distance $\rho \ge 0$ of $e$ does not distinguish one-step targets from later reachable targets.

Proof. Every subsequent path vertex is strictly above $e$, so cannot belong to the antichain. The initial meeting has distance zero.