Equality of the contact-process critical values beyond survival

Let $T$ be an infinite tree, and let $\rho$ be its root. For the contact process on $T$, define

$$\lambda_a=\inf\{\lambda:{\bf P}_\rho(\rho\in\xi(t)\text{ for arbitrarily large }t)>0\},$$ $$\lambda_b=\inf\{\lambda:\limsup_{t\to\infty}{\bf P}_\rho(\rho\in\xi(t))>0\},$$ $$\lambda_2=\inf\{\lambda:\liminf_{t\to\infty}{\bf P}_\rho(\rho\in\xi(t))>0\},$$

and

$$\lambda_c=\inf\{\lambda:\xi(t)\text{ conditioned on not being empty }\mathrel{\stackrel{\mathcal D}{\Rightarrow}}\overline\mu\neq\delta_0\},$$

where $\overline\mu$ is the upper invariant measure and $\delta_0$ is the point mass at the empty set. Equality conjecture. For any infinite tree,

$$\lambda_a=\lambda_b=\lambda_2=\lambda_c.$$

These critical values distinguish increasingly strong forms of persistent infection at the root and complete convergence. The equality is stated as an expected relation for arbitrary infinite trees; the source gives no resolution.