I would use the "Boarding Begins" time on your BP as the deadline to get to the gate. That time usually depends on the aircraft type. The BP also shows the "Boarding Ends" and "Flight Departs" time, like this -
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Not sure if a mobile BP shows this information - another reason I'm still using paper BP's. It would be easier to show to the agent, especially if the "Boarding Ends" time hasn't passed yet.