Let $ A $ be a set of non-zero integers.  Then $A$ contains a sum-free subset $B$ of size $ |B|> \frac{|A|}{3} $ (a result of Erdős).  It is conjectured that RHS can be improved to $\frac{|A|}{3} +10$.  Is there any evidence/heuristic justification for this conjectured bound?

(A set $B$ is *sum-free* if it contains no solution to $x+y=z$.)