Impossibility of exact attainment for the asymptotic rate–error lower bound

An asymptotically scale-invariant centered MRQ is a family of centered multi-resolution quantizers $\{Q_s\}_{s>0}$ whose asymptotic distribution function is denoted by $\overline{F}_{\{Q_s\}_s}$. Let $F_{2^{\mathrm{Unif}[-1,0]}}$ denote the distribution function of $2^U$ for $U\sim\operatorname{Unif}[-1,0]$. Impossibility conjecture. There does not exist an asymptotically scale-invariant centered MRQ $\{Q_s\}_{s>0}$ such that

$$\overline{F}_{\{Q_s\}_s}=F_{2^{\operatorname{Unif}[-1,0]}}.$$

If true, this would show that the lower bound in the rate–error tradeoff cannot be attained exactly, because exact equality is equivalent to the existence of such a family. The conjecture is presented as unresolved in the source.