A scrub jay can remember when it cached a particular piece of food in a particular place, researchers have discovered, and tend not to bother to recover a perishable treat if stored long enough to have rotted.
tend not to bother to recover a perishable treat if
they tend not to bother recovering a perishable treat
tending not to bother to recover a perishable treat it
tends not to bother recovering a perishable treat
tends not bothering to recover a perishable treat it
ING forms aren't verbs, and so don't have a tense of their own. they just adopt the tense of the surrounding clause.
in this case, provided this problem is from a reputable official source, you can take this as proof that "to bother VERBing" is a legitimate idiomatic structure.
in fact, this may be another case in which there are two legitimate structures, but the test goes for the one that gets rid of undesirable repetition.
note that your suggested alternative contains "to" twice in the space of three words: "to bother to recover". that's considered "ugly" -- see here:
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