Folkman's second conjecture for increasing sequences and infinite arithmetic progressions

Let $A$ be an increasing sequence of positive integers, allowing repetitions, and let $A(n)$ denote the number of elements of $A$, counted with multiplicity, that are at most $n$. Folkman's second conjecture. For any sufficiently large constant $C$, if $A(n)\ge Cn$ for all sufficiently large $n$, then the subset-sum set $S_A$ contains an infinite arithmetic progression. The supplied excerpt does not state a resolution of this second conjecture, unlike the first Folkman conjecture, so its status is left open.