No, the investigation was never closed. It was
close to being closed, and it's asserted in the recently public Stzork-Page correspondence that at one point it should have been except for FBI incompetence:
Page
“Phew. But yeah that’s amazing that he is still open. Good, I guess.”
Stzork
"Our utter incompetence actually helps us. 20% of the time, I’m guessing
”
At that point, Stzork messaged the case manager to keep it open, presumably to explore the Logan act angle.
The DOJ claims that the only predicate for the interview was Logan Act violations, and that being a silly reason, invalidates everything after that point and makes 1001 violations immaterial. Opponents argue that the interview was for the larger Russia Collusion investigation. The FBI notes that we've seen from just before the interview are more supporting of the DOJ's record of events:
“What’s our goal? Truth/Admission or to get him to lie, so we can prosecute him or get him fired? If we get him to admit to breaking the Logan Act, give facts to DOJ & have them decide."
...But as far as I understand materiality under 1001 that typically shouldn't matter, even if it looks a bit bad.
What I still don't understand though is why the FBI needed to do the whole runaround of invoking some absurd dusty law which would never succeed at court, and then attempt to get Flynn to lie about it when the FARA violation sounds like a such a slam dunk in and of itself. Flynn took a lot of money from bad people, lobbied for them, and failed to disclose it until after he was discovered. So why were the FBI on--and possibly over excepting "helpful incompetence"--the verge of closing the investigation in the first place?