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I'm definitely not saying welcome them in to fuck up the garden. I agree with getting things in writing and making sure the neighbours take responsibility for any work that needs to be done in the aftermath. Just pointing out that plenty of people think 'it'll be fine' and might take affront at the insinuation there could be damage (taking it personally rather than DH saying "look, I work with scaffolders and they make a mess").
I have a similar situation where the side wall of my house meets the neighbours' drive, which I don't have any rights of access over. The roof is a little tired, and while the neighbours would rather I don't ever step foot on or inconvenience their off-street parking, they also don't want a tile through the windscreen. And the wall needs repointing. Knackered slates falling onto DH garden isn't great either. I can't really tell from the photo whether they could scaffold from front to back (more expensive because longer spans) or whether it's not really possible. Pretty sure there is a general right to access for basic maintenance purposes.
Personally I'd tell the 'neighbours' to fuck off* unless they can show they have some kind of access right. I bet they don't though.
Looking at the map above their proposal strikes me as a nonsense.
* unless, you know, they are actual human beings, but the fact that in their dealing with DH they just rock up and make demands lends me to think they are probably dicks. Do they understand their position? They've built some crappy extension (or is it a garage / studio? IDK) up to the boundary and haven't thought about the practicalities of maintenance and yet they are like 'alright DH you don't know me but can I fuck up your garden?'
(I'm totally grouchy this morning as I've wasted two hours battling a fucking toilet seat of all things. I wouldn't really tell them to fuck off, just ask for more info then ghost them if I didn't like the sound of it / them)