Erdős problem 153

Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $$\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$$ as $\lvert A\rvert\to \infty$?