A conjecture and a prize

Today I would like to discuss a conjecture I recently made in additive combinatorics. I will award £5 for a solution to this conjecture. First, we need some notation.

Let $A$ be a subset of the natrual numbers. Define $A+A=\{a+b \mid a,b \in A\}$. It is a well-studied problem to look at the value of $\lvert A+A \rvert$ compared to $\lvert A \rvert$. My problem concerns when $A$ is made up of nine square numbers.

I have found a set of 9 squares such that $|A+A|=30$. This was achieved, with the help of fellow PhD student George Stagg, by finding three sets of arithmetic progressions of squares of length 3 with the same common difference. My conjecture is that 30 is the smallest value we can obtain for $|A+A|$.

CONJECTURE

Let $A$ be a set of nine square numbers. Then $\lvert A+A \rvert \geq 30$.

If you believe you have a solution, even positive or negative, then please get in touch and claim your £5.

As a final note I will mention that the existence of a perfect cuboid would give a set of nine squares such that $\lvert A+A \rvert = 25$. But no such object has been found and it is itself conjectured not to exist.