Phylogenetic diversity (PD) is a measure of the evolutionary legacy of a group of species, which can be used to define conservation priorities. It has been shown that an important loss of species diversity can sometimes lead to a much less important loss of PD, depending on the topology of the species tree and on the distribution of its branch lengths. However, the rate of decrease of PD strongly depends on the relative depths of the nodes in the tree and on the order in which species become extinct. We introduce a new, sampling-consistent, three-parameter model generating random trees with covarying topology, clades relative depths and clades relative extinction risks. This model can be seen as an extension to Aldous' one parameter splitting model (β, which controls for tree balance) with two additional parameters: a new parameter α quantifying the relation between age and richness of subclades, and a parameter η quantifying the relation between relative abundance and richness of subclades, taken herein as a proxy for overall extinction risk. We show on simulated phylogenies that loss of PD depends on the combined effect of all three parameters, β, α and η. In particular, PD may decrease as fast as species diversity when high extinction risks are clustered within small, old clades, corresponding to a parameter range that we term the `danger zone' (β<-1 or α<0; η>1). Besides, when high extinction risks are clustered within large clades, the loss of PD can be higher in trees that are more balanced (β>0), in contrast to the predictions of earlier studies based on simpler models. We propose a Monte-Carlo algorithm, tested on simulated data, to infer all three parameters. Applying it to a real dataset comprising 120 bird clades (class Aves) with known range sizes, we show that parameter estimates precisely fall close to a danger zone: the combination of their ranking tree shape and non-random extinctions risks makes them prone to a sudden collapse of PD.