The directed LQG metric existence and scaling-limit conjecture

For $\gamma\in(0,2)$ and $\widetilde\theta\in(0,\pi)$, let $\mathfrak L_\Phi^{\operatorname{LDP}}$ and $\mathfrak L_\Phi^{\operatorname{SDP}}$ denote candidate random functions obtained from regularized longest and shortest directed LQG lengths, respectively, for points $z,w\in\mathbb C$. Directed LQG metric conjecture. There exist two distinct random functions $\mathfrak L_\Phi^{\operatorname{LDP}}$ and $\mathfrak L_\Phi^{\operatorname{SDP}}$ with these interpretations. When $\gamma=\sqrt{4/3}$ and $\widetilde\theta=\pi/2$, they describe the scaling limits of longest and shortest directed paths in uniform bipolar-oriented triangulations; moreover, $\mathfrak L_\Phi^{\operatorname{SDP}}$ exists for $\widetilde\theta\in[\pi,2\pi]$ and coincides with the undirected $\gamma$-LQG metric when $\widetilde\theta=2\pi$. This conjecture proposes continuum metrics encoding directed path lengths and their discrete scaling limits, extending the relation between bipolar-oriented maps and LQG beyond the triangulation setting.