Sharpness conjecture for tropical gonality bounds of lattice polygons

Let $\Delta\subset\mathbb{R}^2$ be a two-dimensional lattice polygon. A regular subdivision $\Delta_1,\ldots,\Delta_r$ determines a metric graph $\Gamma(\Delta_1,\ldots,\Delta_r)$, and $U(f)$ denotes the curve associated with a Laurent polynomial $f$. Sharpness conjecture. There is a regular subdivision $\Delta_1,\ldots,\Delta_r$ of $\Delta$ such that the set of irreducible Laurent polynomials $f\in\mathbb{C}[x^{\pm 1},y^{\pm 1}]$ with $\Delta(f)=\Delta$ and with the gonality of $U(f)$ equal to the gonality of $\Gamma(\Delta_1,\ldots,\Delta_r)$ is Zariski dense in the space of Laurent polynomials with $\Delta(f)\subset\Delta$. This conjecture is stated as the expected sharpness of the lower bound obtained from metric graphs and is known in the special case $\Delta=2\Upsilon$; the general assertion remains open.