A counter-example to persistence in generalised preferential attachment trees

Consider a generalised preferential attachment tree with attachment function f, that is a random tree, where at each time-step a node connects to an existing node v with probability proportional to f((v)), where (v) denotes the degree of the node in the existing tree. We provide a counter-example to a conjecture of the author asserting that under the assumption ∑ⱼ₌₁∞ 1/f(j)² < ∞ there is a persistent hub in the model, that is, a single node that has the maximal degree for all but finitely many time-steps. The counter-example is a minor modification of a related counter-example due to Galganov and Ilienko.