A new law aimed at ending the United States’ notorious reputation as a haven for “shell” companies created to obscure crimes will require tens of millions of businesses to report ownership information for the first time, starting Jan. 1.

But there’s a big problem, according to a newly released survey: Awareness among businesses that will have to comply, and even advisers such as certified public accountants and lawyers, is extremely low, raising the specter that business owners will be hammered with hefty fines, potentially even prison time.