Castryck–Cools generic gonality conjecture for lattice polygons

Let $\Delta\subset\mathbb{R}^2$ be a two-dimensional lattice polygon. For a Laurent polynomial $f\in\mathbb{C}[x^{\pm 1},y^{\pm 1}]$, write $\Delta(f)$ for its Newton polygon, and let $U(f)$ denote the corresponding curve in the torus. The lattice width of a polygon is denoted by $\operatorname{lw}(\Delta)$; let $\Sigma$ be the standard $2$-simplex and let $\Upsilon=\operatorname{Conv}\{(-1,-1),(1,0),(0,1)\}$. The sharpest applicable bound is $\operatorname{lw}(\Delta(f))$, except that it is $\operatorname{lw}(\Delta(f))-1$ when $\Delta(f)$ is equivalent to $d\Sigma$ for some $d\geq 2$ or to $2\Upsilon$. Castryck–Cools' generic gonality conjecture. The set of irreducible Laurent polynomials $f$ for which $\Delta(f)=\Delta$ and this sharpest applicable bound is attained is Zariski dense in the space of Laurent polynomials $f$ for which $\Delta(f)\subset\Delta$. The conjecture is proved for all lattice polygons of lattice width at most $4$ and in numerous additional cases; computational evidence supports it up to genus $13$, but the general statement remains open.