Negative-temperature tropical Gibbs-measure conjecture for convex bodies

Let $P\subset\mathbb{R}^{n}$ be a convex body containing $0$ in its interior, and let $R_P\in(0,1]$ be the invariant of $P$ defined in the source. For fixed $\beta<0$, let $\mu_\beta^{(N_k)}$ be the corresponding Gibbs measure on $(\mathbb{C}^{*n})^{N_k}$ when defined. Negative-temperature Gibbs-measure conjecture. For sufficiently large $k$, $\mu_\beta^{(N_k)}$ is a well-defined probability measure if and only if $\beta>-R_P$. If $\beta>-R_P$, then the random empirical measure $\delta_{N_k}$ under $\mu_\beta^{(N_k)}$ converges in law as $N\to\infty$ to

$$\mu_\beta=e^{\beta\psi_\beta}dV,$$

where $\psi_\beta$ is the unique solution in $\mathcal{L}_{P,+}(\mathbb{C}^{*n})$ of the stated Monge–Ampère equation on $\mathbb{C}^{*n}$. The conjecture is motivated by the tropical analogue; the supplied text gives no resolution.