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The relevant implications in the causal hierarchy areThe first follows from strict cone widening: failure of strong causality would give causal curves leaving and re-entering arbitrarily small neighborhoods, and the wider metric turns the limiting recurrence into a closed timelike curve. For the second, if distinct $p,q$ had $I^{+}(p)=I^{+}(q)$, arbitrarily short timelike segments near either point and this equality would produce a causal curve leaving and re-entering a sufficiently small causally convex neighborhood. ThereforeThe stated null geodesic completeness is stronger than needed: stable causality already gives the conclusion.