The converse of the one-dimensional infinite-cluster criterion

Consider the GN$(1,\underline{\alpha})$ model on a Poisson process on $\mathbb{R}$, where $\alpha_i$ are the connection parameters and clusters are formed according to the model's connection rule. Converse of the one-dimensional criterion. If

$$\sum_{i=1}^\infty i\alpha_i<\infty,$$

then all clusters in the GN$(1,\underline{\alpha})$ model are finite almost surely. The preceding proposition proves the complementary statement that divergence of this sum makes every point's connection range infinite; the conjecture would therefore give the expected sharp criterion for percolation in this one-dimensional model.