Consistency-provability strengthening conjecture

Let $\mathcal T$ be the class of theories under consideration, and let $Con_S$ mean that $S$ is consistent. Consistency-provability strengthening conjecture. For every $S,T\in\mathcal T$, if $T$ proves $Con_S$, then the lengths of $S$-proofs of $Con_T(\bar n)$ cannot be bounded by a polynomial in $n$. This proposes a concrete sufficient condition for the relative strength in the finite consistency conjecture.