Folkman's conjecture on square-root-density sequences being subcomplete

Let $A=\{a_1<a_2<a_3<\dots\}$ be an increasing sequence of positive integers, let $A(n)$ count its elements not exceeding $n$, and let

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

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

Folkman's conjecture. There is a constant $c$ such that every increasing sequence satisfying

$$A(n)\geq cn^{1/2}$$

is subcomplete.

This is presented as a refinement of Erdős's completeness conjecture, removing the modularity condition. Folkman proved sufficiency of the stronger bound $A(n)\geq cn^{1/2+\epsilon}$, while the square-root threshold remains the conjectural case in the supplied text.