They're doing a lot better but the various clubs haven't beaten anyone of note yet. Häcken were very lucky to beat their Icelandic opponents in the first round (they scored a very late goal to go through) and with the exception of the Estonian team, all the other teams Swedish clubs have played have been in their off-season.
Helsingborg are slight favourites in my eyes but I don't think there is much between them and Drogheda. And contrary to your suggestion, I don't think the general Swedish football public have a clue about LOI football. And from reading Helsingborg quotes from Stuart Baxter and Henrik Larsson, I don't think Helsingborg know that much either.
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