The tropical Brill–Noether cycle conjecture

Let $\Gamma$ be a tropical curve of genus $g$, and let $W^r_d \subseteq \operatorname{Pic}^d(\Gamma)$ denote the locus of divisor classes of degree $d$ and rank at least $r$. Set

$$\rho=g-(r+1)(g-d+r).$$

Tropical Brill–Noether cycle conjecture. Assume $\rho\geq 0$. Then there exists a canonical tropical subvariety $Z^r_d\subseteq W^r_d$ of pure dimension $\rho$ such that

$$[Z^r_d]=\left(\prod_{i=0}^r\frac{i!}{(g-d+r+i)!}\right)[\Theta]^{g-\rho}$$

modulo tropical homological equivalence. This proposes a tropical analogue of the Brill–Noether cycle formula, giving a canonical cycle inside the rank-$r$, degree-$d$ Brill–Noether locus with the expected dimension and Porteous-type class.