A 52-year-old woman from Delhi presents with acute right upper quadrant pain and fever (38.5°C) for 6 hours. She has a history of recurrent biliary colic over the past 2 years. On examination, Murphy's sign is positive. Ultrasound abdomen shows a 1.8 cm stone in the gallbladder fundus with thickened wall and pericholecystic fluid. During emergency laparoscopic cholecystectomy, the surgeon encounters significant inflammation in the hepatocystic triangle with obscured anatomy and dense adhesions. Which anatomical structure is at highest risk of iatrogenic injury in this scenario?