Tropical Riemann–Roch conjecture for divisors in moderate position

Let $X$ be an $n$-dimensional compact tropical manifold, and let $D$ be a tropical submanifold of codimension $1$ on $X$. Say that $D$ is in moderate position on $X$ when it satisfies the paper's definition of moderate position. For each $k\geq 0$, let $|D^k|$ be the support of the $k$-th power of $D$ in $X$, and let $H^{\bullet}(X\setminus D;\mathbb{R})$ denote the cohomology of the complement.

Moderate-position tropical Riemann–Roch conjecture. If $D$ is in moderate position on $X$, then

$$\operatorname{RR}(X;D)=\sum_{k=0}^{\infty}\chi(|D^k|)=\chi\bigl(H^{\bullet}(X\setminus D;\mathbb{R})\bigr).$$

This is the paper's main conjecture, giving a geometric interpretation of the Riemann–Roch number. It is proved for compact tropical surfaces by the paper's main theorem, and in particular holds for compact tropical surfaces admitting a Delzant face structure; the general higher-dimensional case remains open.