Erdős's minimal-growth conjecture for regular asymptotic bases

Let $h\geqslant 2$ and let $f$ be a function for which the notion of a $\Theta(f)$-regular asymptotic $h$-basis is defined. Adapted Erdős 1956 conjecture. There are no $\Theta(f)$-regular asymptotic $h$-basis with

$$f(x) = o\left(x^{1/h}\log(x)^{1/h}\right).$$

This is presented as a version of Erdős's conjecture concerning the possible growth of representation functions. The preceding discussion gives existence results at the scale $\Theta(x^{1/h}\log(x)^{1/h})$, while the asserted lower-growth exclusion is left unproved.