Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a tropical polynomial. Below we use the following notation: $a \oplus b = \text{max}(a,b)$, and if $\alpha_i=(\alpha_{i1}, \cdots, \alpha_{id})$ then $x^{\alpha_i} = x_1^{\alpha_{i1}} \cdots x_d^{\alpha_{id}}$. $$f(x) = c_1 x^{\alpha_1} \oplus c_2 x^{\alpha_2} \oplus \cdots c_r x^{\alpha_r}$$ The Newton polytope $N(f)$ of a tropical function $f$ is the convex hull of set ${(c_1, \alpha_{1}), \cdots, (c_r, \alpha_{r})}$, where each of these is viewed as a point in $\mathbb{R}^{d+1}$. A linear region of $f$ is a maximal connected subset of $\mathbb{R}$ on which $f$ is linear. It is well known that the number of linear regions of $f$ is equal to the number of vertices in the upper hull of $N(f)$ (see Section 3 of this paper for more details).
Now consider the tropical rational function $f-g$, where $f$ and $g$ are tropical polynomials. Is there an analogous result enumerating the number of linear regions of $f-g$ (for instance, expressed in terms of the Newton polytopes $N(f)$ and $N(g)$)?