The binary analogue of Brauer's addition-chain bound

Let $\|n\|_{2}$ denote the binary complexity of the positive integer $n$. Let $r$ be a positive integer and let $s$ be a nonnegative integer, with $(r,s)\ne(1,0)$. The binary analogue of Brauer's bound. For every $n$ satisfying

$$2^{rs}\leq n<2^{r(s+1)},$$

we have

$$\|n\|_{2}\leq (r+1)s+2^{r}-2.$$

This bound is proposed as an analogue of Brauer's method for establishing the asymptotic formula for addition-chain length; if it holds, it would imply $\|n\|_{2}\sim\log_{2}(n)$.