Opinions please

I_have_spoken

Your deeds may state who is responsible for maintenance of the boundary walls/fences or joint responsibility

A boundary wall could be Party Wall, but even then the law doesn't make repairs liable to 50:50 split.

If the deeds say it's your responsibility to maintain the wall and it falls down causing injury or damage, there could be liability based on duty of care, esp. if it can be shown you were aware of the issue.

DRP

Is it a brick wall?

Presumably the side of the wall with the buttresses will be the owner?

If i know anything about landlords and agents they will want to do it on the cheap - if it was my wall or a shared wall I wouldn't want them touching it at all.

If it is their wall they can do what they want, so it may be to your benefit to contribute and therefore to some extent dictate what is done.