Strict separation of survival and percolation thresholds on lattices

Determine whether the contact process on the integer lattice \(\mathbb Z^d\), for every \(d\ge 2\), satisfies the strict inequality \(\lambda_1(\mathbb Z^d)<\lambda_p(\mathbb Z^d)\), where \(\lambda_1\) is the survival threshold and \(\lambda_p\) is the percolation threshold of the upper invariant infection.

Background

The paper studies the percolation threshold of the upper invariant measure of the contact process and proves the strict inequality $\lambda_1<\lambda_p$ on regular trees. The authors contrast this result with the situation on $\mathbb Z^d$, where finiteness of $\lambda_p$ is known for $d\ge2$, but strict separation from the survival threshold remains unresolved.

The problem asks whether the upper invariant infected set can fail to percolate throughout a nontrivial interval of infection rates above the survival threshold on the integer lattice. Such a result would establish a distinct percolation transition analogous to the one proved in the paper for regular trees.