This is a modification of an unanswered problem on the math StackExchange.

When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?

If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3$, $k=10$. Could there be other integer solutions for $(n,k)$?

Surely the answer is NO, but I am unable to prove that $n=3$, $k=10$ is the only possibility.