On our other car, which has remote start, it will stay running if you open the door.
However, the display in the dash has a "NO KEY!" alert, and if you sit in the car without the fob in your pocket, pushing the brake pedal will kill the car.
The fob security on that car is done with 4 transceivers, one next to cupholders for the PTS (Push To Start), one in each front door, and one at the trunk. Infiniti chose this way because these 4 transceivers have extremely (intentionally) short range. The PTS transceiver near the cupholders basically covers your pockets or a purse/bag that may be set in pass. seat or floor board. The moment my butt rolls out of the driver seat with my fob in my right pocket, the "NO KEY!" alert comes on and the car begins making a chirp before I even get out of the door way.
The fob must be very near any of it's 4 areas before any of those areas are able to be opened. I have tried standing at my driver door and ask my son to unlock the pass door, thinking the presence of my fob would allow it, nope.
So, it can be done. However, in 2020, does it surprise anyone that the remote start is cancelled when a door is opened? Manufacturers now feel compelled to have reminders to "check the back seat" because apparently people are unable to remember their kids are back there on a hot day.