Erdős Problem #39 — Infinite Sidon sets with counting function near $n^{1/2}$

Sidon also asked: Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence for which all the sums $a_i + a_j$ are distinct. Put

$$h(n) = \sum_{a_\ell < n} 1.$$

Probably there is a Sidon sequence $A$ for which

$$h(n) > n^{1/2 - \varepsilon},$$

but (6) is far beyond reach.