Dyadic clock structure conjecture for the perturbed Hofstadter recursion

Let $Q(n)$ be the sequence and define the clock sequences

$$t_1(n)=n-Q(n-1),\qquad t_2(n)=n-Q(n-2).$$

Let $\phi_1$ and $\phi_2$ be bounded or slowly varying functions. Dyadic clock structure conjecture. There appear to exist such functions satisfying

$$t_1(n)=\frac n2+\phi_1(\log_2 n)+o(n),\qquad t_2(n)=\frac n2+\phi_2(\log_2 n)+o(n).$$

Thus the recursive indices are conjectured to remain asymptotically close to $n/2$, with fluctuations governed primarily by the logarithmic scale. The claim is based on numerical experiments and is not resolved in the supplied text.