Euclidean discrete uniformization conjecture for weighted marked surfaces

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$, $\chi(M)=0$, and $d$ is a piecewise linear metric on $(M,V,\varepsilon)$. Let $\mathcal{D}(d)$ denote the discrete conformal class of $d$, and let discrete curvature be the curvature assigned to the marked vertices by the discrete conformal structure. Euclidean discrete uniformization conjecture. There exists a piecewise linear metric $d'\in\mathcal{D}(d)$, unique up to scaling and an isometry homotopic to the identity on $(M,V,\varepsilon)$, such that $d'$ is discrete conformal to $d$ and has discrete curvature $0$. This conjecture predicts a zero-curvature representative in every Euclidean discrete conformal class on a weighted marked surface with Euler characteristic zero; its resolution is not given here.