Truncated tropical second main theorem for meromorphic functions

Assume that $f$ is a nonconstant tropical meromorphic function satisfying

$$\limsup_{r\rightarrow\infty}\frac{\log T_f(r)}{r}=0.$$

Let $a_j$ $(j=1,\ldots,q)$ be distinct values of $\mathbb{TP}^1$, defining tropical linear polynomials $P_j(x)$ on $\mathbb{R}^2$, respectively. If $\lambda=\operatorname{ddg}(\{P_1\circ f,\ldots,P_q\circ f\})$, then, outside an exceptional set of zero upper density measure,

$$(q-2-\lambda)T_f(r)\leq\sum_{j=3}^{q}N^{1)}\left(r,\frac{1_o}{P_j\circ f}\oslash\right)+o(T_f(r)).$$

Let instead $a_j$ $(j=1,\ldots,q)$ be distinct finite values in $\mathbb{R}$ defining tropical linear polynomials $P_j(x)$ on $\mathbb{R}^2$, respectively, with $f\oplus a_j\not\equiv a_j$. Then, outside an exceptional set of zero upper density measure,

$$qT_f(r)=\sum_{j=1}^{q}N^{1)}\left(r,\frac{1_o}{P_j\circ f}\oslash\right)+o(T_f(r)).$$

Truncated tropical second main theorem conjecture. These truncated inequalities and equality are conjectured to hold as stated. They are tropical analogues of the truncated classical Nevanlinna second main theorem; the source explains that the corresponding untruncated tropical results motivate this proposal and that the growth assumption is necessary for related defect statements.