Linear growth conjecture for the perturbed Hofstadter recursion

Let $Q(n)$ denote the sequence defined by the perturbed Hofstadter recursion. Linear growth conjecture. The sequence satisfies

$$\lim_{n\to\infty}\frac{Q(n)}{n}=\frac12.$$

Numerical experiments strongly indicate slope $1/2$, but the supplied text gives no proof or resolution.