The monotone complexity coding conjecture for $P^M$

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

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

This conjecture asks whether the small correction terms in the preceding coding bounds can be removed for the measure $P^M$. The source motivates it by analogy with Shannon–Fano coding and leaves its status unresolved.