The zero-boundary extension of the two-sided local-limit conjecture

Use the notation and hypotheses of the two-sided local-limit conjecture: the rescaled endpoints $z_i^n$ converge to $z_i$, the limiting process $\xi^\star$ is stopped when $m^\star$ reaches the cutoff, and the discrete process is conditioned at time $t_1$.

Zero-boundary extension conjecture. The two-sided local-limit conjecture also holds with the cutoff parameter set to $\epsilon=0$.

This is presented as a further goal after the conjecture with positive cutoff. Its resolution is not independently established by the supplied evidence, so the database status remains open.