Connectedness threshold conjecture for reflected random-walk ranges

Let $Q=\sqrt{({\mathbf{c}}-1)/6}$ and let $X^k$ be the continuous-time random walk on $\mathcal M_k$ reflected off infinity, stopped when it first hits the boundary at time $\tau_k$. Reflected random-walk range connectedness conjecture. The range $X^k([0,\tau_k])\subset\mathcal V\mathcal M_k$ is almost surely connected if $Q>\sqrt{3/2}$, equivalently ${\mathbf{c}}>10$, and has positive probability of being non-connected if $Q<\sqrt{3/2}$, equivalently ${\mathbf{c}}<10$. The threshold is motivated by the Euclidean dimension $3/4$ of Brownian cut points and the conjectural correspondence between LQG singular points and map ends; the behavior at $Q=\sqrt{3/2}$ is left open.