The tropical minimum conjecture for theta-function combinations

Let $\Theta_v$ be the theta functions indexed by the relevant tropical points, and let $a_v$ be non-zero coefficients. For a linear combination $\sum a_v\Theta_v$, write $(\sum a_v\Theta_v)^{\mathrm{trop}}$ for its tropicalization and $\Theta_v^{\mathrm{trop}}$ for the tropicalization of an individual theta function. Tropical minimum conjecture. For any linear combination with non-zero coefficients,

$$\left(\sum a_v\Theta_v\right)^{\mathrm{trop}}=\min \Theta_v^{\mathrm{trop}}.$$

The source notes that one has the inequality $\geq$ a priori and proposes equality as the third basic conjecture needed for the theta-function basis construction. Its resolution status is not given.