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This short note aims at complementing the results of the recent workarXiv:2302.05396, where Jahnel and L\"uchtrath investigate the question ofexistence of a subcritical percolation phase for the annulus-crossingprobabilities in a large class of continuum percolation models, namely theclassical or generalized weight-dependent random connection models. Their workrelates the absence of a subcritical phase to the occurrence of long edges inthe graph, through a somewhat indirect criterium, that stays inconclusive insome models of interest. We provide a more direct and arguably simplercriterium, that is always conclusive when considering classicalweight-dependent random connection models.

Only long edges can erase the subcritical annulus-crossing phase of weight-dependent random connection models