Extremal percolation probabilities among isotropic degree-two models

Let $\mathcal{C}_p$ be the class of probability distributions on $\Omega$ that are independent and identically distributed over the vertices, isotropic, and have expected degree equal to $p$. For $\mathbf{P}\in\mathcal{C}_p$, let $\theta(\mathbf{P})$ denote its percolation probability, and let $\theta^{\text{ns--ew}}_p$ and $\theta^{\text{aon}}_p$ denote the percolation probabilities of the northsouth-eastwest and all-or-nothing models at parameter $p$. Extremal-model conjecture. For every $\mathbf{P}\in\mathcal{C}_p$,

$$\theta^{\text{ns--ew}}_p \geq \theta(\mathbf{P}) \geq \theta^{\text{aon}}_p.$$

This proposes that the northsouth-eastwest and all-or-nothing models are respectively extremal among isotropic, vertex-i.i.d. environments with a fixed expected degree. It is motivated by numerical observations and variance intuition, and remains an open problem.