The tropical Hodge conjecture

Let $\mathfrak X$ be a smooth projective tropical variety. For an integer $p$, consider the tropical cohomology groups

$$H^{p,p}_{\operatorname{trop}}(\mathfrak X,\mathbb{Q})\quad\text{and}\quad H^{p-1,p+1}_{\operatorname{trop}}(\mathfrak X,\mathbb{R}),$$

and let $N$ be the tropical monodromy operator between them. Tropical Hodge conjecture. The locus of Hodge classes in $H^{p,p}_{\operatorname{trop}}(\mathfrak X,\mathbb{Q})$ generated by classes of codimension $p$ tropical cycles in $\mathfrak X$ coincides with the kernel of

$$N\colon H^{p,p}_{\operatorname{trop}}(\mathfrak X,\mathbb{Q})\to H^{p-1,p+1}_{\operatorname{trop}}(\mathfrak X,\mathbb{R}).$$

This is the general tropical analogue of the Hodge conjecture. The paper proves the assertion for rationally triangulable smooth projective tropical varieties; the unrestricted form stated here remains open.