Erdős problem 868

Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) > \epsilon \log n$ for large $n$ and an arbitrary fixed $\epsilon > 0$, then must $A$ contain a minimal additive basis of order $2$?

Larsen and Larsen [LaLa26] constructed a counterexample with $f(n) > c \log n$ for all large $n$.