Marionette lines can be a difficult problem to correct. First, your PS must determine the cause of the Marionette lines, which is actually a depression below the outer part of the lower lip plus "hanging over" of the tissue just outside of the lines (depression).
Fat injections will certainly help the depression, but usually will not eliminate them. You can start with a filler injection in the depression to see if this correction is adequate for you. If it is not, then fat injections will not help any more. If it is, then try the fat injections. It is my opinion that stem cells should be concentrated in the fat to be injected, which should make the fat grafts last longer.
If the correction with the filler is not adequate, the tissue lateral to the Marionette line can be excised, which is a minor procedure. But, it takes several months for the scar to look it best, which is usually an imperceptible scar. This procedure will also elevate the corner of the mouth, eliminating the "down-in-the-mouth look." I like the procedure, but a filler should be tried first before fat injections or any surgical procedure.
Good luck.