If a customer has provided you with false information, your first priority should be in closing the account. SAR filing is a secondary issue at best.
Form 111 places enough emphasis on documentation to make it clear that a filing is required if the appropriate dollar threshold is met; i.e. whether you are you sure of enough information to name a suspect.
Law enforcement's previous knowledge of a situation is irrelevant to whether you file a SAR.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.