Critical-frontier conjecture for the independent model

Let the independent model be the two-parameter percolation model on $\mathbb{Z}^d$ described in the paper, with parameters $(p,\lambda)$, where $d\geq 2$. For $p>0$, let $\lambda_{\mathrm{c}}(p,d)$ denote the conjectured critical value in the $\lambda$-parameter.

Independent-model phase-diagram conjecture. There exists $\lambda_{\mathrm{c}}(p,d)\in(0,1)$ such that $\lambda_{\mathrm{c}}(\mathord{\cdot},d)$ is continuous and strictly increasing on $(0,1]$; for $\lambda<\lambda_{\mathrm{c}}(p,d)$ there exists almost surely no infinite blue cluster; and for $\lambda>\lambda_{\mathrm{c}}(p,d)$ there exists almost surely a unique infinite blue cluster. Moreover, for each fixed $\lambda$, the probability that the origin lies in an infinite blue cluster is non-increasing in $p$.

The paper proves several regions of the independent-model phase diagram, including regions with and without a unique infinite blue cluster, but the complete critical frontier and the stated monotonicity properties are not established.