3SUM Hard Problems

Let $A$, $B$, and $C$ be finite sets of integers whose total cardinality is $n\in\mathbb{N}$. The 3SUM decision problem asks whether there exist $a\in A$, $b\in B$, and $c\in C$ such that

$$a+b=c.$$

A problem is 3SUM-hard in the sense used by the source when this decision problem reduces to it. Determine whether 3SUM-hard problems can be solved in $O(n^{2-\epsilon})$ time for some fixed $\epsilon>0$, including in expectation.