Relative tropical Riemann–Roch conjecture for a pair of divisors

Let $X$ be a compact $n$-dimensional tropical manifold. Let $D$ and $D'$ be tropical submanifolds of codimension $1$ on $X$, or empty. Assume that $D'$ is in moderate position on $X$, that the restriction $D'|_D$ is a tropical submanifold of $D$ with support $D'\cap D$, and that $D'\cap D$ is in moderate position on $D$. For $k\geq 0$, let $|D'^k|$ be the support of the $k$-th power of $D'$ in $X$, and let $|(D'|_D)^k|$ be the corresponding support on $D$. Let $H^{\bullet}(X\setminus D',D\setminus D';\mathbb{R})$ denote the relative cohomology of the pair.

Relative tropical Riemann–Roch conjecture.

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

and this equals

$$\chi(X\setminus D')-\chi\bigl(D\setminus(D'\cap D)\bigr).$$

This extends the conjectures for effective and anti-effective divisors by incorporating a divisor difference and relative cohomology. The source does not provide a resolution in general, so the higher-dimensional and unrestricted cases remain open.