Bucić–Hendrey–Mohar–Steiner–Yepremyan linear perimeter-gap conjecture

Let the perimeter gap of a digraph be the difference between its number of vertices and the length of its longest directed cycle. Bucić–Hendrey–Mohar–Steiner–Yepremyan's conjecture. There exists an $\varepsilon>0$ and infinitely many values of $n$ for which there exists a connected vertex-transitive digraph on $n$ vertices whose perimeter gap is at least $\varepsilon n$. This would strengthen their known logarithmic lower bound on the perimeter gap to a linear one; the conjecture remains open.