Universal-theory Herbrand consistency conjecture

Let $T\in\mathcal T$ be a theory axiomatized by a universal sentence, and let $HCS(T)$ denote its Herbrand Consistency Search problem. Universal-theory Herbrand consistency conjecture. The theory $T$ does not prove that $HCS(T)$ is total for any formalization of it by a $\Delta^b_1$ formula. This proposes a natural candidate for a search problem outside the problems whose totality is provable in $T$.