Universal convergence bound for MDL predictions of Bernoulli sequences

Let $\theta_0$ be the true Bernoulli parameter, let $\theta^x$ denote the MDL prediction based on the observed sequence, let $K(\theta_0)$ be the prefix Kolmogorov complexity of $\theta_0$, and let $\Delta(k)$ denote the relevant complexity difference. Universal convergence bound. The cumulative expected squared prediction error should satisfy

$$\sum_n {\mathbf E} (\theta_0-\theta^x)^2\stackrel\times\leq K(\theta_0)+ \sum_k 2^{-\Delta(k)}.$$

This would strengthen the preceding upper-bound theorem in the universal setting, where the complexity is prefix Kolmogorov complexity, and would imply the paper's earlier convergence result up to a constant. The source does not establish the bound.