It's a Gibson, so once you pop the headstock off, it will easily fit in the overhead compartment.
One option would be to get a quality flight case and use that for the whole trip back.
If the guitar comes with a Gibson hard case and you want to keep that instead, I would fly with the guitar in its case (headstock supported) and packed double boxed like you were going to ship it. Detuned and something between the strings and the fingerboard. Just check the box as an oversized checked box. It will be pricey ($100-150), but you will keep the original hard case clean and damage free when you arrive. Use quality packing tape to make a "carry handle" for the box as you or the airport guys lug it around.
The stress of hoping you may be able to carry on a nice guitar is not worth it IMHO. They are supposed to let you carry on an instrument, but I think you are at the mercy of what ever gate attendant you have the day you fly.
I don't fly with guitars any more, but if I did have to check one, I may consider popping in an Apple AirTag just in case the box didn't arrive with me.