Comparison of \(\lambda_2\) and the healthy-vertex percolation threshold

Determine whether the strong-survival threshold \(\lambda_2\) is strictly smaller than the upper-invariant healthy-vertex percolation threshold \(\lambda_{p^\complement}\) for the \(d\)-regular tree when \(3\le d\le6\).

Background

The paper proves $\lambda_2<\lambda_{p^\complement}$ for $d\ge7$, where $\lambda_{p^\complement}$ is the threshold beyond which the healthy vertices no longer percolate under the upper invariant measure.

The corresponding comparison in degrees three through six is not established. A positive answer would extend the ordering of the survival and healthy-set percolation transitions to all regular trees considered in the paper.