Tropical Severi irreducibility conjecture for toric surfaces

Let $S$ be a toric surface over an algebraically closed field of characteristic zero, let ${\mathcal L}\in \operatorname{Pic}(S)$ be an effective class, and let $g\geq 0$ be a non-negative integer. Let $V^{\operatorname{irr}}(S,{\mathcal L},g)$ denote the Severi variety parameterizing irreducible nodal curves of genus $g$ in the linear system $|{\mathcal L}|$ that do not contain the zero-dimensional orbits of $S$. Severi irreducibility conjecture. The variety $V^{\operatorname{irr}}(S,{\mathcal L},g)$ is either empty or irreducible. The conjecture predicts irreducibility of the relevant Severi varieties for toric surfaces in characteristic zero; the source recalls it as a conjecture from earlier work, while the supplied material gives no resolution.