Tropicalization conjecture for non-Archimedean lattices

Let $K$ be the non-Archimedean field and let $L\subset K^n$ be a lattice. Write $\varphi_L$ for its entropy polynomial, $\Sigma_L$ for the associated polyhedral complex, and $V_L:=L\otimes_{\mathcal O_K}K$. For $\alpha>0$, define

$$Q_{L,\alpha}(x)=\exp\big(-\alpha\varphi_L(x)\big),\qquad x\in\mathbb R^n.$$

Tropicalization conjecture. For every lattice $L\subset K^n$, there exists $\alpha_L\in\mathbb R_{>0}$ such that, for every $\alpha\geq\alpha_L$, $Q_{L,\alpha}$ is the survival function of a probability measure on $\mathbb R^n$, and this measure is supported on $\Sigma_L$. In particular, its projection to $\mathbb R^n/\mathbb R\bm{1}_n$ is a probability measure on $\operatorname{trop}(\mathbb P(V_L))$. This conjecture proposes a probabilistic realization of the tropicalization associated with a non-Archimedean lattice; the statement is presented as a motivation following the known supermodularity of entropy vectors of lattices.