What is the conjecture

https://www.erdosproblems.com/43

If $A,B\subset {1,\ldots,N}$ are two Sidon sets such that $(A-A)\cap(B-B)={0}$ then is it true that
$$\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1) $$,
where $f(N)$ is the maximum possible size of a Sidon set in ${1,\ldots,N}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound be improved to

$$\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq (1-c)\binom{f(N)}{2}$$
for some constant $c>0$?

Status: open

Prerequisites needed

None, Sidon sets is already in ForMathlib