They review passport stamps with the entries in your Physical Presence Calculator that was submitted with your application. Everyone in my test cohort went through the process and they actually review every stamp.
The officers have access to systems that allow them to verify your entry/exit but then that is an extra step after the fact. During my interview they had questions about some of my US stamps based on the visa classification. So I did not go DM immediately but a few days later.
My recommendation is to carry your passport if you can arrange for it to be shipped to you else just try your luck. You can explain it and see where it leads