Fast-speed comparison conjecture for contact-process critical values

Assume the leading parametrisation of the dynamical random tree model, with parameters $\alpha$, $\sigma$, and $\eta$, and let $\lambda_1,\lambda_2$ be its critical values and $\lambda_1^p,\lambda_2^p$ those of the corresponding penalised contact process. Suppose

$$\frac{1}{2}<\alpha<1\wedge\frac{1}{2\sigma}\quad\text{and}\quad\alpha+2\eta\geq 1.$$

Fast-speed comparison conjecture. Under these assumptions, $\lambda_1>0$ if and only if $\lambda_1^p>0$, and $\lambda_2>0$ if and only if $\lambda_2^p>0$.

The conjecture asserts that, in this part of the fast-speed regime, the dynamical model and the penalised contact process have the same positivity criteria for both critical values. The passage notes that existing results establish this comparison only for some parameter choices, leaving the full stated region open.