Separation of survival and root-reinfection thresholds on strongly exponentially growing trees

Let $T$ be an infinite tree. Call a vertex $v$ essential if $T\setminus\{v\}$ has at least two infinite components. Say that $T$ has strongly exponential growth if there is some integer $r\geq1$ such that, for every $v\in T$, there are at least three essential vertices $w$ at graph distance $r$ from $v$. Let $\lambda_1$ be the survival threshold and let $\lambda_a$ be the threshold above which the root is reinfected at arbitrarily large times; together with $\lambda_b$, $\lambda_2$, and $\lambda_c$, these are the critical values defined for the contact process above. Separation conjecture. For any tree with strongly exponential growth, either all the above critical values are zero, or

$$\lambda_1\neq\lambda_a.$$

The claim concerns whether survival can begin strictly before recurrent reinfection of the root on sufficiently branching trees. The source gives no resolution.