Continuity of the contact-process survival probability on trees

Let $T$ be an infinite tree, and consider the contact process on $T$ started from a single infection. Write $\lambda$ for the infection rate and let the survival probability be the probability that the process never becomes empty. Continuity conjecture. For any tree, the survival probability starting from a single infection is a continuous function of $\lambda$. The conjecture concerns the nature of the first phase transition, namely the transition at the survival threshold $\lambda_1$. The source gives no resolution.