Is there any circulant $(-1,1)$-matrix satisfying the following property?

Let $n$ be a positive integer greater than $13$, is there any n-by-n circulant $(-1,1)$-matrix $A$ satisfying the following property:

$$AA^T=(n-1)I+J$$

where $I$ is the n-by-n identity matrix and $J$ is the n-by-n matrix of ones.

I conjecture that the answer is no. But I can't prove it.