Betti-number conjecture for lattice polygons and toric surfaces

Let $\Delta\subseteq\mathbb{R}^2$ be a lattice polygon whose interior polygon $\Delta^{(1)}$ is two-dimensional and contains $g\geq 4$ lattice points. Assume that $\Delta^{(1)}\not\cong\Upsilon$. Denote the graded Betti table of $X_{\Delta^{(1)}}\subseteq\mathbb{P}^{g-1}$ by $b_\ell$ as in the source. The toric Betti-table conjecture. Then

$$\min\{\ell\mid b_{g-\ell}\neq 0\}=\begin{cases} \operatorname{lw}(\Delta^{(1)})+1 & \text{if }\Delta^{(1)}\cong(d-3)\Sigma\text{ for some }d\geq 5,\\ \operatorname{lw}(\Delta^{(1)})+1 & \text{if }\Delta^{(1)}\cong2\Upsilon,\\ \operatorname{lw}(\Delta^{(1)})+2 & \text{in all other cases.} \end{cases}$$

The conjecture predicts the first relevant nonzero Betti number of the toric surface from the lattice width of its interior polygon, with two explicitly identified exceptional families. The supplied material gives no resolution status.