Determine the limiting shortest-path metric as gamma approaches two

Determine whether an interesting limiting directed metric exists for the shortest-path directed LQG metrics as gamma tends to 2 with theta fixed in (0, pi), and characterize its limiting behavior if it exists.

Background

The paper discusses a conjectural critical limit of the longest-path directed LQG metric as gamma tends to 2 and relates it to Brownian separable permutons. The authors explicitly state that they are less certain about the shortest-path metric and leave the existence of a nontrivial limiting object unresolved.

References

We expect that, at least for $\theta \in (0,\pi)$, the scaling exponent converges to $1/2$ as $\gamma \to 2$ with $\theta$ fixed, but we are not sure whether there is an interesting limiting directed metric.

— Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits  (2510.26123 - Borga et al., 30 Oct 2025) in Remark 3.4, Section 2.3, Directed LQG metrics