Peak location law for the perturbed Hofstadter recursion

Let $F(m)$ be the frequency sequence and let

$$B_k=\{2^k,\dots,2^{k+1}-1\}.$$

For each block, let $m_k$ be an index at which $F(m)$ attains its maximum. Peak location law. The peak locations satisfy

$$m_k=\frac{4}{3}2^k+O(1).$$

Numerical experiments suggest this geometric scaling, but the statement is not established in the supplied text.