I think Clifford has the better argument by far.

Her opposition brief is very strong. She points out that Cohen has already run his mouth extensively about the hush money transaction in the media, so his interest in shutting up can’t be that strong. She points out that Cohen himself initiated an arbitration proceeding to try to shut Clifford up. She points out that Cohen filed a declaration in this civil case about the formation of the contract even after the FBI’s search warrant, and gave a statement about the matter the day after the search.

Moreover, Clifford points out that Cohen’s motion is extremely vague — it amounts to saying “look, the FBI is investigating the same general subject.” He doesn’t explain what specific subjects of testimony (including, for instance, the source of funds or his communications with Trump) that could incriminate him. Finally, she points out that the case is of national interest and has attracted huge public attention, weighing against a stay.

Cohen’s failure to commit that he will take the Fifth doesn’t help either. Perhaps because taking the Fifth is so widely perceived as a sign of guilt (which, of course, it is not), Cohen’s not ready to say definitely that he will take the Fifth to any questions about the transaction — he’s sticking with the wishy-washy argument that his rights are “implicated.” That’s compounded by the fact that his lawyers, asked about him taking the Fifth, publicly denied that he had decided to do so, and suggested that doing so was premature. In other words, they undercut their own stay request.