Nonuniform reflection incompleteness conjecture

Let $\mathcal T$ be the class of theories under consideration, and let $\Sigma^b_1RFN_T(\bar n)$ denote the finite $\Sigma^b_1$ reflection principle for $T$. Nonuniform reflection incompleteness conjecture. For every $S\in\mathcal T$, there exists $T\in\mathcal T$ such that the lengths of $S$-proofs of $\Sigma^b_1RFN_T(\bar n)$ cannot be bounded by $p(n)$ for any polynomial $p$. This is the reflection-theoretic analogue of the nonuniform finite consistency conjecture.