The tropical Ehrhart coefficient bound by tropical minors

Let $\mathbb{T}\mathbb{N}$ denote the tropical natural numbers, let $M \in \mathbb{T}\mathbb{N}^{d \times m}$, and let $P=\operatorname{tconv}(M)$ be the corresponding tropical lattice polytope. For $i\in[d]=\{1,\ldots,d\}$, let $c^b_i(P)$ be the $i$th tropical Ehrhart coefficient and let $\operatorname{tm}_i(M)$ be the $i$th tropical minor. The tropical minor conjecture. For every $i\in[d]$,

$$\operatorname{Log}\,\operatorname{card}\{c^b_i(P)\}\leq \operatorname{tm}_i(M).$$

This conjecture proposes that the maximal tropical $i$-minors provide upper bounds for the corresponding tropical Ehrhart coefficients, extending the analogous bound established earlier in the paper. The case $i=1$ is proved by the preceding proposition, while the general bound is left as a conjecture.