An addition chain is a finite sequence of positive integers that starts at $$1$$, so that any element of the sequence is a sum of two previous elements. That is, it is a sequence $$(a_1, \ldots, a_k)$$ where $$a_1 = 1$$, and for each $$i > 1$$ there are $$j, k < i$$ such that $$a_i = a_j + a_k$$. An example of an addition chain is $$(1, 2, 3, 6, 12, 15).$$ We call an addition chain that ends in $$k$$ an addition chain for $$k$$. We write $$l(k)$$ for the length of the shorted addition chain for $$k$$, not counting the 1 at the start of the sequence. Our example above shows that $$l(15) \leq 5$$; in fact $$l(15) = 5$$.
Write $$C(k)$$ for the set of addition chains for $$k$$, and $$s(c)$$ for the length of a chain $$c$$ (again not counting the initial 1). Furthermore, we use the notation $$c_1 \cup c_2$$ informally to mean joining the chains $$c_1, c_2$$ while pruning repeat elements. Then we trivially have $$l(k) = \min_{i < k \\c_1 \in C(i)\\ c_2 \in C(k - i)} s(c_1 \cup c_2) + 1.$$ Now in many cases, we will find that this minimum is not achieved if we restrict to $$c_1, c_2$$ such that $$s(c_1) = l(i)$$ and $$s(c_2) = l(k - i)$$; to achieve maximal overlap, we may want a sequence for one of the summands which is suboptimal. My question is:
Is it possible that we need both subsequences to be suboptimal? That is, is the same minimum attained if we restrict to pairs $$(c_1, c_2)$$ such that at least one of $$s(c_1) = l(i)$$ and $$s(c_2) = l(k - i)$$ holds?