A 48-year-old woman from Mumbai undergoes a left superficial parotidectomy for a benign mixed tumor. During surgery, the surgeon carefully identifies the facial nerve trunk at the stylomastoid foramen and preserves all five terminal branches. However, 2 weeks postoperatively, the patient develops weakness of the lower lip with inability to depress the lower lip and pull it downward and laterally. The remaining facial nerve branches function normally. Which nerve branch was likely injured during the procedure?