Monotonicity and critical-limit conjecture for the divide and color model

Let $\mathcal{L}\in\{\mathbb{Z}^2,\mathbb{H}\}$ be either the square or hexagonal lattice, let $p_c^{\mathcal{L}}$ denote its critical percolation parameter, and let $r_c^{\mathcal{L}}(p)$ denote the critical value in the divide and color model at parameter $p$. Monotonicity and critical-limit conjecture. On the interval $p\in[0,p_c^{\mathcal{L}})$, the function $r_c^{\mathcal{L}}(p)$ is strictly decreasing, and

$$\lim_{p\to p_c^{\mathcal{L}}-}r_c^{\mathcal{L}}(p)=\frac{1}{2}.$$

The conjecture describes the behavior suggested by the simulations for the square and hexagonal lattices; its strict monotonicity and limiting value are not established by the rigorous confidence-interval method presented in the paper.