Erdős Problem #870

Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ (where $r(n)$ counts representations of $n$ as a sum of at most $k$ elements of $A$) then $A$ must contain a minimal basis of order $k$? The claimed answer is no, for every $k\geq 3$.