Hi 7Q,
I know I'm not much help but there does seem to be a difference between the requirements of Rishworth (for the LGW-based crews) and those of OSM (for the FLL crew). The LGW requirements do not mention narrow body in longhaul operations. From the Rishworth brief:
Minimum experience required
Commander: 6,000 hours total time, 2,500 PIC hours on any EFIS commercial jet aircraft. 1,000 hours on a wide body aircraft.
Current type rating in EASA license.
Given this, I would suggest that no PIC on a wide body is required but, no doubt, I'm sure they would prefer that over narrow body experience.
I think only the agencies can answer your specific question (which I think is a good one), so why not give them a quick call/email and ask them directly.