Linear perimeter-gap conjecture for vertex-transitive digraphs

For a directed graph, its circumference is the maximum length of a directed cycle, and its perimeter gap is the difference between its order and its circumference. Linear perimeter-gap conjecture. There exists an $\varepsilon>0$ and infinitely many values of $n$ for which there is a connected vertex-transitive digraph of order $n$ whose perimeter gap is at least $\varepsilon n$. The paper proves a logarithmic lower bound and identifies this linear bound as a more ambitious intermediate target; the supplied source does not indicate whether the conjecture has been resolved.