Immunisation conjecture for fast degree penalisation

Assume the leading parametrisation of the dynamical random tree model, with $\alpha$ the degree-penalisation exponent, $\kappa$ the degree parameter, $\eta$ the speed parameter, and $\nu_2$ the update parameter. Immunisation means that the critical value satisfies $\lambda_1=\infty$.

Immunisation conjecture. If $\alpha\geq 1$, then for every sufficiently small $\kappa>0$ there exists $\nu'=\nu'(\kappa,\eta)$ such that

$$\lambda_1=\infty$$

for all $\nu_2<\nu'$.

The conjecture predicts an immunisation phase when the degree-penalisation exponent is at least one and the update parameter is sufficiently small. It is motivated by the absence of exceptionally large effective degrees in this regime; the passage gives no resolution.