Depends on the court. Some conflate/join the 4th and 5th, most separate them.
The argument for separation is that ordering you to submit to a search is not the same as compelling you to be a witness against yourself.
-- If I say "I refuse to answer on the grounds that it may incriminate me", do you think they have the right to imprison me until I reveal the key? --
The court has the power to do it, even if the court is wrong.
The prosecutor's argument will be that you have to open the door to your house pursuant to a warrant, even if the contents of your house may incriminate you.
The prosecutor can go fuck himself because I don't have to open the door just because they flash a warrant. They are free to knock the door down if they are able. But I am not compelled to help.