The uniqueness conjecture for universal perfect state transfer among circulants with property $\mathbb{T}$

A complex unit gain graph is a graph whose adjacency matrix has complex entries, with each nonzero entry of unit magnitude. A graph has property $\mathbb{T}$ if all nonzero coefficients in its adjacency matrix are complex numbers with unit magnitude. The notation $\operatorname{Circ}(0,-\mathtt{i},\mathtt{i})$ denotes the circulant graph specified by the first row $(0,-\mathtt{i},\mathtt{i})$. Uniqueness conjecture. $\operatorname{Circ}(0,-\mathtt{i},\mathtt{i})$ is the only circulant with property $\mathbb{T}$ which has universal perfect state transfer. This conjecture proposes a complete classification of circulant graphs with property $\mathbb{T}$ exhibiting universal perfect state transfer; the preceding discussion establishes uniqueness among graphs on three vertices, while the general case remains open.