Riemann–Roch conjecture for tropical surfaces

Let $\Delta$ be a tropical surface, let $D$ be a Cartier divisor on $\Delta$, and let $K_\Delta$ be the divisor obtained by summing $(\operatorname{deg} r-2)[r]$ over the ridges $r$ of $\Delta$. Assume that $K_\Delta$ is also a Cartier divisor. Write $\chi(\Delta)$ for the Euler characteristic of the underlying topological space of $\Delta$. Riemann–Roch conjecture. One has

$$h^0(\Delta,D)+h^0(\Delta,K_\Delta-D)\geq \frac{\operatorname{deg}D\cdot(D-K_\Delta)}{2}+\chi(\Delta).$$

This is a formal analogue of the Riemann–Roch theorem for algebraic surfaces, using $h^0(\Delta,K_\Delta-D)$ as a substitute for top cohomology in the absence of a higher-cohomology theory. The source states the result conjecturally and gives no evidence of a resolution.