Finite-reinforcement percolation conjecture for planar reinforced 2-out percolation

Consider the finite-$\alpha$ reinforced $k$-out percolation model on $\mathbb Z^d$, and let $\mathcal E_\infty$ be the set of edges that are eventually reinforced in every round. In the limiting case $\alpha=\infty$, this is the corresponding infinitely reinforced-edge configuration.

Finite-reinforcement percolation conjecture. Assume that $d=k=2$. Then there exists $\alpha_0>0$ such that

$$\mathbb P\big(\mathcal E_\infty\text{ percolates}\big)=1$$

whenever $\alpha>\alpha_0$.

This conjecture extends the limiting-model percolation conjecture to sufficiently large but finite reinforcement exponents. The paper presents it as plausible but does not prove it, so its resolution remains open.