I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome.
I managed to prove that (assuming $S$ is the connected subset) $S$ minus one of the three small triangles (but don't remove the two points which connect the small triangle to the rest) is also connected. This can be stated more generally for connected subsets in three spaces cyclically connected by three points. (basically, if $S$ restricted to one of them fails to be connected, then the restriction to the other ones has to be connected)
I'm not sure how to continue. Additionally, $S$ doesn't have to be closed, which causes further problems. Any ideas are also welcome.