The multiple-free subset-sum conjecture for powers of two

Let $z\in\mathbb{N}$ and let $A=\{a_1,\ldots,a_k\}\subseteq[2^z]$, where $[2^z]=\{1,2,\ldots,2^z\}$. For each non-empty subset $A_i$ of $A$, write $S(A_i)$ for the sum of its elements. Multiple-free subset-sum conjecture. If for every $i,j\le 2^k-1$ the quotient

$$\frac{S(A_i)}{S(A_j)}$$

is not a power of $2$, then

$$k=z+O\left(1\right).$$

The condition says that the subset sums form a multiple-free sequence with respect to powers of $2$. The conjecture is motivated by numerical data and would sharpen the known lower-bound constructions for this problem; the paper does not provide a proof or resolution.