Erdős's lower-bound conjecture for representation functions of asymptotic bases

Let $h\geqslant 2$, let $\mathcal{S}$ be the space of subsets of the nonnegative integers, and for an asymptotic $h$-basis $A$ let $r_h(n;A)$ denote its representation function. Erdős's conjecture. If $A\in\mathcal{S}$ is an asymptotic $h$-basis, then

$$\limsup_{n\to +\infty} \frac{r_h(n;A)}{\log(n)} > 0.$$

This is stronger than the Erdős–Turán conjecture because it asserts growth on the scale of $\log(n)$ rather than merely unboundedness. The source discusses the limitations of probabilistic methods and gives no proof or counterexample.