Likely they would win a tribunal case if you took them to court due to past precedents. But do it if you want, you might be able to get the amount you have to pay reduced. Keep in mind the following, if you lose, they will reclaim their legal expenses from yourself. If you want to challenge it, give these people a call, they give free telephone consultations so you could briefly explain the situation and then they can tell you if you have a chance at a tribunal to either reduce your share or not pay at all.
leasehold-law.com – find the phone number and get a telephone consult
You might have a chance at reducing it as that is a ridiculous amount of money. The lifts in our building was like 3300 per leaseholder flat. We have to pay despite the stupid things not going to our floor just the way it is in a rip off leasehold building. Seems like a good buy when the flat is less than most in the area particularly if you get the mega local authority tenant discount but the ongoing charges will bankrupt the avg person.