No written test involved. Obtaining a FAA PPL issued on the basis of a foreign licence (https://www.law.cornell.edu/cfr/text/14/61.75) really is just about paperwork, patience (a lot of it) and then showing up at your local FSDO upon arriving in the States for a quick interview before they issue your temporary.

Here's where to get the ball rolling on the FAA side:
https://www.faa.gov/licenses_certifi..._verification/

You can also reach out to the Airmen Certification Branch in Oklahoma via phone for foreign certificates related questions. They have always been very helpful to me.