Combinatorial tropical gonality conjecture for lattice polygons

Let $\Delta\subset\mathbb{R}^2$ be a two-dimensional lattice polygon, and let $\Delta^{(1)}$ denote the convex hull of the interior lattice points of $\Delta$. Write $\operatorname{lw}(\Delta^{(1)})$ for its lattice width, let $\Upsilon=\operatorname{Conv}\{(-1,-1),(1,0),(0,1)\}$, and let $\Gamma(\Delta_1,\ldots,\Delta_r)$ be the metric graph associated with a regular subdivision $\Delta_1,\ldots,\Delta_r$ of $\Delta$. Combinatorial tropical gonality conjecture. There exists a regular subdivision $\Delta_1,\ldots,\Delta_r$ of $\Delta$ such that the gonality of $\Gamma(\Delta_1,\ldots,\Delta_r)$ equals $\operatorname{lw}(\Delta^{(1)})+2$ if $\Delta\not\cong 2\Upsilon$, and equals $3$ if $\Delta\cong 2\Upsilon$. This combines the generic-gonality and lower-bound conjectures into a purely combinatorial statement; it is not established in general.