The essential skeleton conjecture for log canonical pairs and tropicalization

Let $(X,\Delta)$ be a log canonical pair such that $U:= X \mathbin{\rule[0.2em]{0.67em}{0.12em}} \Delta$ is a subvariety of a torus. Let

$$\textsf{trop}:U\longrightarrow\mathscr{T}U$$

be the tropicalization map, and let $\textsf{Sk}^{\textsf{ess}}(X,\Delta)$ denote the essential skeleton. Essential skeleton conjecture. The restriction

$$\textsf{trop}|_{\textsf{Sk}^{\textsf{ess}}(X,\Delta)}:\textsf{Sk}^{\textsf{ess}}(X,\Delta)\longrightarrow\mathscr{T}U$$

is surjective with finite fibers. The theorem for $(\overline{\textsf{M}}_{0,n},\partial\overline{\textsf{M}}_{0,n})$ described above is presented as a special case. The statement proposes a general relationship between birational geometry, essential skeleta, and tropicalization; no resolution of the general conjecture is given here.