Demailly (2012) showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents; Moreover, the statement that all strongly positive currents with rational cohomology class can be approximated by positive linear combinations of integration currents, can be viewed as a strong version of the Hodge conjecture (1982). In this talk, I will explain the construction of a current which does not verify the latter statement on a toric variety, where the Hodge conjecture is known to hold. The example belongs to the family of `tropical currents', which we extend their framework to toric varieties, discuss their extremality properties, and express their cohomology classes as recession fans of their underlying tropical varieties. Finally, the counter-example will be the tropical current associated to a 2-dimensional balanced subfan of a 4-dimensional toric variety, whose intersection form does not have the right signature in terms of the Hodge index theorem. This is a joint work with June Huh.