Conjectural characterization of tropical characteristic polynomial coefficients for symmetric matrices

Let $A$ be a symmetric tropical matrix of size $n$, and let $(\delta_0,\delta_1,\ldots,\delta_n)$ be the coefficient sequence of its tropical characteristic polynomial. For $3\leqslant m\leqslant n$, at least one of the following inequalities holds:

$$\delta_m-\delta_{m-1}\leqslant\delta_{m-1}-\delta_{m-2},$$

or

$$\delta_m-\delta_{m-2}\leqslant2(\delta_{m-2}-\delta_{m-3}).$$

Conjectural characterization. For every $m$ with $3\leqslant m\leqslant n$, at least one of these two inequalities holds. The inequalities express a concavity-type regularity in the coefficient sequence of the tropical characteristic polynomial of a symmetric matrix.

The preceding results establish analogous necessary inequalities for the first coefficients, while an example shows that these conditions need not hold for nonsymmetric matrices. Extending the observed pattern to all coefficients is proposed as a direction for future research.