IMHO it must be a licensed airfield.
The reason is that the lapsed pilot has the same ratings as a 20 hour solo student (none) and is being sent solo under the exact same requirements as a solo student pilot. i.e. they have a medical, fly from a licensed airfield and are supervised by an instructor.
Remember that some countries require student pilots to hold student licenses. If that was the case here then this lapsed pilot would also have to hold a valid student licence also.
I believe that a lapsed rating is gone and one can not use it for anything including avoiding the requirements for training at a licensed airfield.
I can't see why a pilot in this case would want to fly solo - Dual training as required followed by a flight test would surely be the full requirement.
Regards,
DFC