Kolmogorov-random-string consistency proof-size conjecture

Let $R$ be the set of Kolmogorov-random binary strings defined using the fixed universal Turing machine $U$, and let $k$ be the threshold supplied by Chaitin's incompleteness theorem for the theory $\mathcal{S}$. For $x\in R$, let $Con_{\mathcal{S}+“x\in R"}(n)$ denote the bounded consistency statement for the extension of $\mathcal{S}$ by the sentence encoding $x\in R$. Kolmogorov-random-string proof-size conjecture. Given $\mathcal{S}$, there exists $\epsilon$ such that, for all $x\in R$ with $|x|\geq k$ and for all $n$, $\mathcal{S}$ requires $2^{n(1-\epsilon)}$ symbols to prove $Con_{\mathcal{S}+“x\in R"}(n)$.