Folkman's conjecture on dense sequences and subcompleteness

Let $A=\{a_1\leq a_2\leq a_3\leq\dots\}$ be an infinite non-decreasing sequence of positive integers, and let $A(n)$ count its elements between $1$ and $n$, with repetitions counted. Its finite subset sums are

$$S_A=\left\{\sum_{x\in B}x:\ B\subset A,\ |B|<\infty\right\}.$$

The sequence is subcomplete when $S_A$ contains an infinite arithmetic progression.

Folkman's conjecture. There is a constant $C$ such that, if $A(n)\geq Cn$ for all sufficiently large $n$, then $A$ is subcomplete.

The conjecture isolates linear density as a sufficient condition for an infinite arithmetic progression in the subset sums. The paper states that its application settles this conjecture.