Existence of graphs of every prescribed minimum girth without the AOP property

Let $g\in\mathbb{N}$. A graph has the AOP property if it admits an acyclic orientation with at most one directed path between any pair of vertices, and its girth is the length of its shortest cycle. High-girth AOP conjecture. For every $g\in\mathbb{N}$, there exist graphs with girth greater than or equal to $g$ that do not have the AOP property. The conjecture challenges the expectation that sufficiently large girth forces the AOP property; the supplied text gives no resolution.