Non-singleton intersection conjecture for pairs of Liouville quantum gravity geodesics

Fix $\gamma\in(0,2)$, a whole-plane Gaussian free field $h$, and distinct points $z_1,z_2,z_3,z_4\in\mathbb{C}$. Let $\Gamma_{z_i,z_j}$ denote a geodesic from $z_i$ to $z_j$ in the metric $D_h$. Non-singleton intersection conjecture. Almost surely,

$$\Gamma_{z_1,z_2}\cap\Gamma_{z_3,z_4}$$

cannot be a singleton. The analogous statement is not known for $\gamma$-LQG and appears to be unknown even for Brownian geometry; it would rule out a type of isolated intersection relevant to the classification of geodesic networks and the multiplicity of their points.