TFNP non-completeness conjecture

Let $\mathcal T$ be the class of theories under consideration. A TFNP problem is a total polynomial search problem, and $\operatorname{TFNP}(T)$ is the class of such problems provably total in $T$; let $\operatorname{TFNP}^*(T)$ be the problems polynomially reducible to members of $\operatorname{TFNP}(T)$. TFNP non-completeness conjecture. For every $T\in\mathcal T$, there exists a TFNP problem $P$ that is not polynomially reducible to any TFNP problem provably total in $T$, equivalently $\operatorname{TFNP}^*(T)\ne\operatorname{TFNP}$. This conjecture seeks a total polynomial search problem whose complexity is not captured, even up to polynomial reduction, by the problems provably total in any fixed theory $T$.