The asymptotic existence conjecture for irreducible non-metrizable path systems

Let $G(n,1/2)$ be the Erdős–Rényi random graph on $n$ vertices, with each edge present independently with probability $1/2$. An irreducible non-metrizable path system is a path system that is both irreducible and non-metrizable.

Asymptotic existence conjecture. Asymptotically almost every graph has an irreducible non-metrizable path system; specifically, this holds with probability

$$1-o_n(1)$$

for $G(n,1/2)$ graphs.

This conjecture asks whether the explicit constructions of irreducible non-metrizable path systems extend to almost all large random graphs. The source presents it as an open problem, and no resolution is supplied here.