Monotonicity of the Ronkin function along the Mahler flow

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial, and let $R_k(x,y)$ denote its Ronkin function at flow parameter $k>0$. For $k_2>k_1$, distinguish the non-linear, bounded linear, and unbounded linear regions of the amoeba complement for $k_2$. Ronkin monotonicity conjecture. For every fixed $(x,y)$ and $k_2>k_1$,

$$R_{k_2}(x,y)>R_{k_1}(x,y)$$

when $(x,y)$ lies in a non-linear or bounded linear region for $k_2$, while

$$R_{k_2}(x,y)=R_{k_1}(x,y)$$

when $(x,y)$ lies in an unbounded linear facet for $k_2$. For $k_2>k_1\geq k_\textup{iso}$, the non-linear region is the liquid phase and a bounded (unbounded) facet is a gas (solid) phase. This extends the proposed monotonicity of Mahler measure to the Ronkin function; the source presents it as an extension without a proof.