Continuity of the complex additive-martingale limit on the phase boundary

Prove that, almost surely, the map z\mapsto W_\infty(z) is continuous on P_1\cup P_{1,2}.

Background

Inside P_1, the limit W_\infty(z) is analytic almost surely, while on P_{1,2} the additive martingale also has a nontrivial limit. The paper identifies continuity across this boundary as an unresolved issue and expects the interior limit to agree continuously with the boundary values.

References

Almost surely, the function $z \mapsto W_\infty(z)$ is continuous on $P_1 \cup P_{1,2}$.

— Fluctuations of additive martingale limits of branching Brownian motion  (2609.10530 - Chen et al., 9 Sep 2026) in Conjecture 2, Section 3, “Related literature and further questions,” paragraph “Complex additive martingales”