The 27–29 geodesic networks conjecture for Liouville quantum gravity

Fix $\gamma \in (0,2)$ and consider $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Let $\operatorname{Star}_k$ denote the set of points joined to the origin by at least $k$ geodesics, and let the network graphs be the geodesic networks under consideration. Geodesic networks conjecture. There are either $27$, $28$, or $29$ network graphs that appear in $\gamma$-LQG: the $27$ networks that appear in the directed landscape, together with the network in Figure (a) if $\dim(\operatorname{Star}_4) \ge 1$ and the network in Figure (b) if $\dim(\operatorname{Star}_4) \ge 3/2$. Geodesic networks occur with dimensions given by the formulas referenced in the source. This is a speculative prediction based on star-dimension heuristics; the required star dimensions are not known for general $\gamma$-LQG, so the claim remains open.