Existence conjecture for divisible design Cayley digraphs with prime-power parameters

Let $n$ be an odd prime power. A divisible design Cayley digraph is a Cayley digraph that forms a divisible design with parameters $(v,k,\lambda_1,\lambda_2,m,n)$, where the parameters specify its order, valency, within-class and between-class common-neighbour counts, and the number and size of the classes. Existence conjecture. There exists a divisible design Cayley digraph with parameters

$$(n^2-1,n,0,1,n+1,n-1).$$

The construction is known in the cases checked by computer, and the conjecture asks whether it exists for every odd prime power $n$.