Definable Tietze extension conjecture for DCULOAS structures

Let $M$ be a definably complete uniformly locally o-minimal structure of the second kind, abbreviated DCULOAS, and suppose that $M$ enjoys the definable Tietze extension property. Definable Tietze extension conjecture. The structure $M$ is o-minimal. The theorem established in the surrounding discussion proves this implication only for archimedean DCULOAS structures; the conjecture remains open in general.