Hi, I don't come into this section very often, but I saw your post and recognised you from the first time when our babies were in the NICU at about the same time!
My smaller twin Joshua was diagnosed with CLD. At the time I was told that the only reason he was diagnosed with it and not his twin, was because he was still oxygen dependent at a certain gestation (can't remember what that was now!) Anyway, Joshua was still on O2 at 36 week gestation, less than your little one, but still resolutely refusing to wean off it. Coming home on O2 was also mentioned to us, and we were told that they would give him 2 more weeks and then they would start to make the arrangements for that to happen. I remember posting on here at the time as the diagnosis of CLD scared me to death (as I made the huge mistake of googling it) as did the thought of coming home on O2.
About a week later one of the nurses suggested to the doctor that Joshua be put on dieuretics to help remove any fluid from his lungs, and lo and behold shortly afterwards he started to wean and relatively quickly too. Both boys (Ben had stayed in to wait for his brother) came home just under a week before their due date & Joshua was only off the O2 completely for 4 days before he came home.
As for the CLD, Josh is now 3 and has not suffered any breathing difficulties since coming home and so I don't think any lasting damage was done to his lung, and in any event as he has grown it has become such a small area of his lungs that it doesn't affect him, and so I scared myself quite unnecessarily with all the horror stories on google!
Jude still has plenty of time to come off the O2 before he is "term" but if he doesn't then my friend has just taken her little girl home on O2 (rather different story as she was term but had severe muconium aspiration) and she isn't finding the O2 at home to be a problem at all and the O2 canister fits comfortably under her pushchair so getting out & about is not a problem.
I hope this helps a little bit & that Jude is soon home x x