Sublinear fluctuations conjecture for the perturbed Hofstadter recursion

Let $E(n)$ denote the fluctuation term in the decomposition of $Q(n)$ into its main term and fluctuation. Sublinear fluctuations conjecture. The fluctuation term satisfies

$$E(n)=o(n).$$

This is presented as an open problem; more precise bounds on the growth of $E(n)$ remain unknown.