Gonality of outerplanar graphs

Determine whether the gonality of outerplanar graphs on n vertices is bounded by O(√n).

Background

Scramble number is studied because it provides a lower bound for graph gonality, and the paper computes square-root-scale scramble numbers for fan and wheel graphs. The authors explicitly leave unresolved whether the gonality itself satisfies the same asymptotic upper bound throughout the class of outerplanar graphs.