Dyadic frequency law for the perturbed Hofstadter recursion

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

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

Dyadic frequency law. The frequencies $F(m)$ satisfy

$$\#\{m\in B_k:F(m)=r\}=2^{k-r+2}\qquad(3\le r\le k+1),$$

together with

$$\#\{m\in B_k:F(m)=k+2\}=1,\qquad \#\{m\in B_k:F(m)=k+3\}=1.$$

The law is suggested by numerical data for dyadic blocks through $k=15$, but no proof or resolution is supplied.