Full monodromy conjecture for rational curves on smooth complete toric surfaces

Let $X$ be a smooth complete toric surface, let $\mathcal{L}$ be a line bundle in the setting of the monodromy map, and let $C$ be a curve whose nodes form the set on which monodromy acts. The monodromy map is denoted by $\mu_\mathcal{L}$. Full monodromy conjecture. The image of the monodromy map $\mu_\mathcal{L}$ is the full permutation group on the set of nodes of $C$. This asserts that all permutations of the nodes occur as monodromy for rational curves on smooth complete toric surfaces; the supplied text does not establish whether the statement is resolved.