Universality conjecture for the dimension of three-stars in Liouville quantum gravity

Fix a whole-plane Gaussian free field $h$ and let $D_h$ be the associated $\gamma$-Liouville quantum gravity metric for $\gamma\in(0,2)$. Let $\mathfrak{S}_3\subseteq\mathbb{C}$ denote the set of points from which three geodesics emanate pairwise disjointly except at their common starting point. Universality conjecture for three-stars. Almost surely,

$$\dim_{D_h}(\mathfrak{S}_3)=2.$$

The precise Hausdorff dimensions of $k$-stars are known in some other random geometric models, including Brownian geometry and the directed landscape, but remain largely unknown for general $\gamma$-LQG. This conjecture predicts that the dimension of the set of three-stars is universal across the LQG parameter range.