Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that

$$\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}$$

for all $\epsilon>0$ and large $N$?