Let $n>1$ be an integer, and $a>1$ a real number. Consider the subspace $L<R^{2^n}$ generated by the $n$ possible tensor products of the $n-1$ copies of the vector $(1,a)$ and one copy of $(a,-1)$. (The $n$ generators of $L$ correspond to the $n$ positions where the factor $(a,-1)$ can be inserted into the product. Say, if $n=2$, then $L$ is generated by the two vectors $(a,-1,a^2,-a)$ and $(a,a^2,-1,-a)$.) Now for a vector $l\in L$ and a positive integer $k\le 2^n$, choose arbitrarily $k$ coordinates of $l$ and let $\sigma$ denote their sum. Since $\sigma$ is the scalar product of $l$ and a vector of norm $\sqrt k$, we have $$ |\sigma| \le \|l\| \sqrt k. $$

My question is whether this trivial estimate can be improved by a growing factor; say,

Is it true that for any $l\in L$ and $k\le 2^n$, the sum of any $k$ coordinates of $l$ is at most $C_a\|l\| \sqrt k / \log\log n$ in absolute value, with a constant $C_a$ depending only on $a$?