Interesting problem and one I’ve faced many times travelling around the world over the past 40 odd years. My experience has been that it depends 99.9% on the single person accepting your luggage plus which class of seat you have purchased. I live in Australia, so going anywhere else is always a long haul flight. I’ve lost count of the many ‘round the world flights’ I’ve done but I must say I’ve been treated very well by most airlines. United can be ‘iffy” at best as this company appears to choose its crews from flight attendants who’s demeanour can be very unpredictable. You need to exercise some keen skills of persuasion. Having said that I’ve nearly always managed to get my guitar on board. My guitar usually being a Strat or a Tele which can usually fit in a large aircraft overhead. I’ve flown with an acoustic but in a flight case and in with the luggage. I once travelled with a young Japanese guy who had a cello with him. He bought it a seat and it sat next to him all the way from London via Singapore to Sydney. Very expensive! I once purchased a new George Harrison Gretsch Country Gent from a shop in Chicago who shipped it to a friends house in LA where I was staying. I was travelling back to Sydney on QANTAS business class. I checked my suitcase in, but took my backpack and the guitar onto the aircraft and was greeted at the door by a very tall lady of colour who asked me what I had in the case. I told her it was a George Harrison guitar. She immediately smiled, turned to a male flight attendant and said very seriously “Hey! This is Mr **** and he has George Harrison in his case. Please look after them both” Well “George Harrison” was very gently placed into the suit locker opposite my seat and we travelled together very comfortably and we’re very well looked after all the way back home to Australia. He still resides comfortably in my music room. I really wish you well, but even if you phone ahead and get the OK from the airline, I doubt you’ll get it in writing and you will almost certainly be at the mercy of that single person at the aircraft door. Good luck with the United Airlines flight crew. Just remember to smile, be very friendly and nice. It might just work.