Geometricity conjecture for morphisms of 1-dimensional tropical fans

Let $X$ and $Y$ be 1-dimensional tropical fans in $\boldsymbol{R}^n$ and $\boldsymbol{R}^m$, respectively. Let $\varphi_Y$ and $\varphi_X$ be their associated semiring homomorphisms, and let $\nu:\varphi_Y\to\varphi_X$ be a morphism.

Geometricity conjecture. Every morphism $\nu:\varphi_Y\to\varphi_X$ is geometric.

The source presents this as sufficient for proving that the functor $\Phi$ is full, using the preceding proposition characterizing morphisms induced by maps of tropical fans. The supplied status is unknown.