Contraction conjecture for shortest homology basis loops in planar metric graphs

Let $X$ be a planar metric graph, meaning a geodesic space homeomorphic to a finite $1$-dimensional simplicial complex that admits a planar embedding. Let $G$ be a coefficient group, and let $\alpha\subset X$ be a simple closed loop such that its homology class $[\alpha]$ is a member of a shortest homology basis of $H_1(X;G)$. A contraction $X\to\alpha$ is a homotopy through maps from $X$ to itself from the identity map to a retraction onto $\alpha$. Planar metric graph contraction conjecture. There exists a contraction $X\to\alpha$. The conjecture asks whether planarity makes the condition that $[\alpha]$ belong to a shortest homology basis sufficient for contracting the entire metric graph onto $\alpha$; the preceding nonplanar examples show that the analogous assertion fails without planarity.