Let $k\geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A\subset\mathbb{N}$ has $\lvert A\rvert=n$ and $\| 1_A\ast 1_A\|_\infty \leq k$ then $A$ contains a Sidon set of size at least $r$.

Is it true that $H_k(n)/n^{1/2}\to \infty$? Or even $H_k(n) > n^{1/2+c}$ for some constant $c>0$?