Tropical Martens' conjecture for metric graphs

Let $\Gamma$ be a metric graph of genus $g$. For integers $d$ and $r$ satisfying

$$0<2r\leq d<g,$$

let $w^r_d(\Gamma)$ denote the Brill–Noether rank, namely the largest integer $\rho$ such that every effective divisor of degree $r+\rho$ is contained in a divisor of degree $d$ and rank $r$. Tropical Martens' conjecture. One has $w^r_d(\Gamma)\leq d-2r$, and equality holds precisely exactly when $C$ is hyperelliptic. The conjecture proposes that Martens' theorem extends to tropical geometry; the preceding proposition establishes the inequality and equality for hyperelliptic graphs, while the converse characterization is the speculative part.