Erdős Problem #44 — Extending every Sidon set to a near-maximal Sidon set

An old problem of mine states as follows: Let $a_1 < a_2 < \cdots < a_k$ be a Sidon sequence. Can one extend it to a larger Sidon sequence

$$a_1 < \cdots < a_k < a_{k+1} < \cdots < a_\ell, \qquad a_\ell = (1 + o(1))\ell^2.$$

In other words, loosely speaking: can one extend every Sidon sequence to a Sidon sequence which is substantially maximal? Many generalizations are possible.