Establish monotonicity of gonality for general circulant graphs

Determine whether the gonality of the circulant graphs \ci_n(J) is nondecreasing with respect to n; specifically, establish or refute the inequality \gon(\ci_n(J))\leq\gon(\ci_{n+1}(J)) for a fixed adjacency list J whenever both circulant graphs are defined and connected.

Background

The paper notes that graph gonality can behave badly under graph operations, so the inclusion of a circulant graph on n vertices into the analogous circulant graph on n+1 vertices does not automatically imply monotonicity of gonality. The authors explicitly state that the relevant inequality is not known in general. They then prove a monotonicity result for even-regular Harary graphs by deleting a vertex from a suitable vertex-transitive graph and pairing its neighbors, yielding the inequality \gon(H_{k,n})\leq\gon(H_{k,n+1}) for even k. This specialized result does not resolve the question for arbitrary fixed-adjacency-list circulant graphs.