This is the case in mine, and I am sure I've read on other forums etc that it is expected behaviour.
Ignoring individual mode for now, let's say you're driving in normal, then you change to sport. The steering gets heavier, the suspension firms up (if you have DCC), and the amount of time gears are held changes, which is indicated by the S on the dash cluster (S1, S2, S3 etc) rather than a D (D1, D2, D3...)
Now, turn the car off and on again, and the car, according to the mode select screen (and the indicator as to what mode you're in that's in the top left of the central screen) will still say sport. The suspension will still be firm, and the steering will still be heavy, BUT the "engine" has reverted to normal again, ie. you get D1, D2, D3 on the dash cluster. Opening the mode select screen and tapping "Sport" again (even if it's already highlighted) reverts the shifts to sport mode, but that aspect definitely starts up in normal, regardless of what mode you were in when turning the car off.
I hope that has cleared things up?