The conjecture on the order types of initial segments of addition chain defects

Let $\ell(n)$ denote the length of a shortest addition chain for the natural number $n$, and define the addition chain defect by

$$\delta^\ell(n):=\ell(n)-\log_2 n.$$

Let

$$\mathscr{D}^\ell=\{\delta^\ell(n):n\in\mathbb{N}\}.$$

Order-type conjecture for addition chain defects. For each whole number $k$, the set $\mathscr{D}^\ell\cap[0,k]$ has order type $\omega^k$.

The set of addition chain defects is known to be well ordered with order type $\omega^\omega$. This conjecture specifies the order type of each initial segment and is described as saying that the limit of the initial $\omega^k$ addition chain defects is equal to $k$.