The strengthened universal MDL convergence bound

Let $K$ be prefix Kolmogorov complexity, let $\boldsymbol{\vartheta}_0$ be the true parameter, and let $\Delta(k)$ denote the complexity difference used in the MDL setup. Strengthened universal convergence conjecture. Under the same conditions as the preceding upper-bound theorem,

$$\sum_n \mathbf{E}(\boldsymbol{\vartheta}_0-\boldsymbol{\vartheta}^x)^2\stackrel{\times}{\leq} K(\boldsymbol{\vartheta}_0)+\sum_k2^{-\Delta(k)}.$$

The conjectured strengthening would make the universal setup useful despite the divergence of $\sum_k2^{-K(k)}\sqrt{K(k)}$, and would imply the paper's earlier convergence result up to a multiplicative constant. The supplied text gives no resolution status.