Will the call drop after the 15-minutes?

No, Meet sessions/Intro calls are scheduled for 15 minutes in order to give the Seeker and the Guide the opportunity to make sure the chemistry is there. But, there is a 10 minute grace period at the end of every session so that you have some 'wrap up' time. 

After that, we unceremoniously drop the call at the end of the 10 minute grace period, which can be very jarring. We're looking into ways to make that a little friendlier as it feels kind of... Spartan