Herbrand consistency search form of the TFNP conjecture

Let $\mathcal T$ be the class of theories under consideration. For a consistent universal sentence $\Phi$, let $HCS(\Phi)$ be its Herbrand Consistency Search problem: given finitely many tuples of terms, find a truth assignment to the relevant atomic subformulas making the corresponding instances of $\Phi$ true. Herbrand consistency search TFNP conjecture. For every $T\in\mathcal T$, there exists a consistent universal sentence $\Phi$ such that $HCS(\Phi)$ is not polynomially reducible to any TFNP problem provably total in $T$, equivalently $HCS(\Phi)\notin\operatorname{TFNP}^*(T)$. This is an equivalent Herbrand-search formulation of the TFNP conjecture.