4 problems

Let $\Gamma$ be a uniform $\operatorname{CLE}_4$ in the unit disk, and consider the conformally invariant metric on $\Gamma\cup\{\partial\mathbb{D}\}$ whose distance from a loop to…

No local extrema conjecture for the modulus metric. The function $\mu(x)=\mu_D(x_0,x)$ cannot have a relative minimum or a relative maximum at any point $x\in D$ with $x\ne x_0$.

Smoothness conjecture for $\mu_D$-minimizers. Every $\mu_D$-minimizer $\gamma_D(x,y)$ is a smooth Jordan arc joining $x$ and $y$.

Rigidity conjecture. An affirmative answer should hold to the question of whether equality in either … implies $g=g_0$.