Jensen–Len tropical Martens conjecture for metric graphs

Let $\Gamma$ be a metric graph of genus $g$, and let $d,r$ be integers satisfying $0<2r\leq d<g$. The Brill–Noether rank $w_d^r(\Gamma)$ is the tropical Brill–Noether rank. Jensen–Len tropical Martens conjecture. One has

$$w_d^r(\Gamma)\leq d-2r,$$

and equality holds precisely when $\Gamma$ is hyperelliptic. This conjecture seeks a tropical analogue of Martens' theorem using Brill–Noether rank rather than the dimension of the Brill–Noether locus; the preceding proposition establishes the inequality and equality for hyperelliptic graphs, while the asserted converse is the conjectural part.