  We prove the absence of percolation in a directed Poisson-based random
geometric graph with out-degree $1$. This graph is an anisotropic variant of a
line-segment based lilypond model obtained from an asymmetric growth protocol,
which has been proposed by Daley and Last. In order to exclude backward
percolation, one may proceed as in the lilypond model of growing disks and
apply the mass-transport principle. Concerning the proof of the absence of
forward percolation, we present a novel argument that is based on the method of
sprinkling.
