They will do but only within a certain time.
If the key was not nearby it would show a message on the dash "key not present"
Bloke on the FRSOC lent his Mk2 FRS to his missus, she dropped him off at work but he still had the key. When she went to park she realised she didnt have it but was able to start the car again as she not long switched it off.
Thats true, but you don`t have long time to do that.