Exponential proof-size conjecture for unprovable consistency extensions

Let $\mathcal{S}$ be a theory, let $\phi$ be a sentence, and let $n$ be a natural number. Write $Con_{\mathcal{S}+\phi}(n)$ for the bounded consistency statement for $\mathcal{S}+\phi$, and say that $\mathcal{S}$ simulates $\mathcal{S}+\phi$ when it has the simulation property described in the paper. Exponential proof-size conjecture. Given $\mathcal{S}$, there exists $\epsilon>0$ such that, for every $\phi$ and $n$, if $\mathcal{S}$ does not simulate $\mathcal{S}+\phi$, then $\mathcal{S}$ requires $2^{n(1-\epsilon)}$ symbols to prove $Con_{\mathcal{S}+\phi}(n)$.