Let $n\ge13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property:

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

where $I$ is the $n\times n$ identity matrix and $J$ is the $n\times n$ matrix of ones.

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