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I'm going to suggest they pay for it themselves and then failing that claim against their insurance.
My main worry is that they got a plumber in and the only damp they found was in the washiing machine which is only a year old. I'm almost certain that it's the pipes rather than the machine but they would not let the plumber cut the back of the cupboards out to check the pipe work...
I think it'll just keep happening
Sorry, looks like a stressful situation to be in. Not an expert, but could you get your house insurance to talk to them or their insurance? Might do the trick