Fragility at two distinct mirror probabilities

Determine whether there exists a connected 4-regular multigraph embedded in the plane, with every compact set meeting only finitely many edges, that is p-fragile for mirrors for two different values of p.

Background

The paper introduces p-fragility for mirrors as the property that the mirror model has infinite trajectories and, almost surely on the event of an infinite trajectory, infinitely many pivotal vertices at which changing the local state makes the trajectory finite. For medial graphs of bundled one-ended trees, the paper proves that mirror fragility is equivalent to Bernoulli bond-percolation fragility and consequently requires p to equal the critical percolation threshold of the underlying bundled tree.

The authors note that this argument uses the monotonicity of Bernoulli percolation in p, whereas mirror configurations do not possess an analogous monotonicity. This leaves unresolved whether a single connected plane 4-regular multigraph can exhibit mirror fragility at two distinct values of p.