Monotonicity conjecture for the process of 1's

Consider two systems on the same lattice dimension $d$ with the same number of types $n$, having parameter vectors $(p_1,p_2,\ldots,p_n)$ and $(\tilde{p}_1,\tilde{p}_2,\ldots,\tilde{p}_n)$. Suppose that $p_1\leq\tilde{p}_1$ and $p_i\geq\tilde{p}_i$ for every $i\geq 2$. Monotonicity conjecture. The process of $1$'s in the first system is stochastically smaller than the process of $1$'s in the second system. The proposed monotonicity result would imply the equal-maximal-rate poisoning conjecture, although the paper notes that it is not necessary and that other monotonicity arguments might also suffice.