Since the aircraft carries a "N" registration I guess the pilot would be subject to to the FAA's definition of "solo" flight, which is contained in 14 CFR 61.51.
eCFR ? Code of Federal Regulations
(d) Logging of solo flight time. Except for a student pilot performing the duties of pilot in command of an airship requiring more than one pilot flight crewmember, a pilot may log as solo flight time only that flight time when the pilot is the sole occupant of the aircraft.
(e) Logging pilot-in-command flight time. (1) A sport, recreational, private, commercial, or airline transport pilot may log pilot in command flight time for flights-
(i) When the pilot is the sole manipulator of the controls of an aircraft for which the pilot is rated, or has sport pilot privileges for that category and class of aircraft, if the aircraft class rating is appropriate;
(ii) When the pilot is the sole occupant in the aircraft;
(iv) When the pilot performs the duties of pilot in command while under the supervision of a qualified pilot in command