OPEN

For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1\not\in A+A$.

There may even be $\gg \lvert A\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.