"to have forgotten" vs "to forget"

Hello,

Would you please help me understand whether variant b) is right as well as a) and what is the difference between them two?

a) I asked them to check everything once again because I don't want them to forget something.

b) I asked them to check everything once again because I don't want them to have forgotten something.

I guess b), unlike a), makes it more clear to the interlocutor that the concern is about that people in question might forget something before some event in future, say, start of the equipment test, rather than in general future time, maybe even after that planned event. It must be isn't that much necessary to use in speech, but just for some emphasis. Am I right?