Density conjecture for sums of distinct powers

Let $s\ge1$ and let $A$ be a sequence of integers at least $2$. For a sequence $S$ of positive integers, write

$$\Sigma(S)=\left\{\sum_{i=1}^{\infty}\varepsilon_i s_i:\ \varepsilon_i\in\{0,1\},\ \sum_{i=1}^{\infty}\varepsilon_i<\infty\right\},$$

and let $\operatorname{Pow}(A;s)$ be the sequence of powers $a^k$ with $a\in A$ and $k\ge s$.

Density conjecture for sums of distinct powers. If every pair $a_1,a_2\in A$ satisfies $\gcd\{a_1,a_2\}=1$ and

$$\sum_{a\in A}\frac1{\log a}>\log 2,$$

\nthen $\Sigma(\operatorname{Pow}(A;s))$ has positive lower asymptotic density.

The question follows a result constructing complete power sequences even when the reciprocal sum $\sum_{a\in A}1/(a-1)$ is arbitrarily small. The conjectured positive-density conclusion is not established in the source.