NP-completeness conjecture for metric divisorial gonality

Fix a positive integer $r$. For a metric graph $\Gamma=(G,l)$ with rational edge lengths, the $r^{th}$ Metric Divisorial Gonality problem asks whether $\operatorname{dgon}_r(\Gamma)\leq k$ for a given integer $k$. Metric divisorial gonality conjecture. The $r^{th}$ Metric Divisorial Gonality problem is NP-complete. Membership in NP is known when both $r$ and $k$ are fixed; extending the argument to variable $k$, for example with $k$ bounded by $r|V(G)|$, is left open.