Circulant unistochastic matrices and circulant unitary matrices conjecture

Let $\mathcal{C}_d$ denote the set of circulant matrices, $\mathcal{U}_d$ the set of unistochastic matrices, and $\mathcal{C}_d^2$ the set of doubly circulant unistochastic matrices. The paper establishes that $\mathcal{C}_d\cap\mathcal{U}_d=\mathcal{C}_d^2$ for dimensions $d\leq 4$. Circulant unistochastic matrices conjecture. The sets $\mathcal{C}_d\cap\mathcal{U}_d$ and $\mathcal{C}_d^2$ coincide for all $d$; equivalently, every circulant unistochastic matrix has a corresponding circulant unitary matrix. This extends the result proved in dimensions at most four and remains open in general.