The generalized probability coding conjecture for $P^G$

Let $B^{\sharp}$ be the domain of finite binary strings under consideration, and let $P^G(x)$ and $K^G(x)$ denote the corresponding generalized probability and complexity functions. Write $KP^G(x)$ for the prefix complexity associated with $P^G$. Generalized probability coding conjecture. For every $x\in B^{\sharp}$ with $P^G(x)>0$,

$$K^G(x)\leq KP^G(x)+O(1).$$

This conjecture asks for the same tight coding inequality in the generalized setting, where the preceding discussion notes that $P^G$ is a semimeasure rather than a measure. Its status is unresolved in the supplied source.