Shared rain water gulley leaking - how approach this with neighbours?

[Iamthinking]

We have a mansard roof in a victorian terraced house. The water gulley that is about a foot wide is just underneath the top bedroom's window. This takes the rainwater run off from both of our neighbours' roofs as well as our own as we have the only downpipe.
(I hope this makes sense).

We have recurrent problems with this leaking into our bedroom below this gulley, but roofers cannot do any proper job to seal it as there is a join to the left and right with the neighbours. If they have badly maintained gulleys (LHS does) then that would mean water can get down underneath our zinc. We would only be putting our zinc on top of whatever crap they have. If we all three got our gulleys recovered in zinc then that would be ideal as we could have a smooth line of three with a good drop either side. But I can't force them to do that.

My ideal scenario is to brick up the gulley to left and right and get them to have their own downpipe, so removing this problem forever. It will require us all getting scaffolding and so being a not inconsiderable expense. But I don't want to sour relations as we get on with them. How do I start the conversation?

It is apparently called a valley gutter..

No, a parapet gutter

Have a chat with LHS about the issues you are having and tell them you've been advised that clearing the whole run is likely to make a considerable difference to the leak. Ask them to have their side cleared and see what the response is. If you can, invite the round so you can show them the damage.

Would retro fitted rain chains help too ?! Might ease the load ?!

OK I have talked to LHS, I bumped into her in the street. They have leaks too and standing water, so I didn't feel I was encouraging them to spend money they didn't have.. It went well, and subject to her speaking to her OH we may be able to get a good gradient on one long run of zinc/lead.

Sorry for such an intensely boring thread!