Planar graph NP-hardness conjecture for higher divisorial gonality

For a fixed positive integer $r$, let the $r^{th}$ Divisorial Gonality problem ask whether a graph has divisorial gonality at most a given integer, and let the $r^{th}$ Stable Divisorial Gonality problem ask the analogous question after allowing subdivisions. Planar higher-gonality hardness conjecture. The $r^{th}$ Divisorial Gonality and $r^{th}$ Stable Divisorial Gonality remain NP-hard when the inputs are restricted to planar graphs. The preceding results establish NP-hardness for bipartite and apex graphs; the planar case is left as a conjecture.