From memory I left it blank and it allowed me through - albeit with the requirement. I put my Marriott Amex number into the application to apply as an existing cardholder. Was good as I got a better than public sign up offer. I don’t recall using private mode in my browser to get it.
I wasn't able to progress through the form without entering a SSN/ITIN, so I called, explained this to the agent and asked to apply using a passport number.
Applying over the phone took 25 minutes as many terms and conditions had to be read out.
When the application was submitted, the outcome was that more information was required.
I was offered the option to call the number of the department needing more information later on, or to be transferred.
I was transferred and just needed to confirm year of birth and the USA mailing address where the card was to be sent to. Then got confirmation of approval!