  We consider some problems related to the truncation question in long-range
percolation. It is given probabilities that certain long-range oriented bonds
are open; assuming that this probabilities are not summable, we ask if the
probability of percolation is positive when we truncate the graph, disallowing
bonds of range above a possibly large but finite threshold. This question is
still open if the set of vertices is $\Z^2$. We give some conditions in which
the answer is affirmative. One of these results generalize the previous result
in [Alves, Hil\'ario, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972
(2017)].
