Log-periodic modulation conjecture for the perturbed Hofstadter recursion

Let $Q(n)$ be the sequence, let $\log_2 n$ denote the base-two logarithm, and let $\rho(n)$ be a smaller correction term. Log-periodic modulation conjecture. There appears to exist a bounded or slowly varying function $\Phi$, approximately periodic with period $1$, such that

$$Q(n)=\frac n2+\Phi(\log_2 n)+\rho(n).$$

The precise form of $\Phi$ is explicitly described as an open problem.