Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \vdots & \ddots & a & b\\ b & \cdots & b & a\\ \end{pmatrix},\ where\ b>0. $$ Does it follow that $A$ is a circulant matrix?

Note: There is a partial classification of non-negative normal matrices posted here, which seems like it can be used to attack this problem.

There is a geometric interpretation as well: both the set of rows and the set of columns of $A$ form equidistant sets of vectors on a sphere, and basic geometry appears to severely restrict the possibilities.