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I would review the arrangement you have with the agents and double check the terms and conditions of the letting agreement you have in place with them. If there is no reference to a payment for ending the contract, I would stand firm on this matter and not pay them.
Should they continue, I would consider reporting the agent to the property ombudsmen for breach of their duty and due diligence - https://www.tpos.co.uk/
I hope this helps and please do let me know if I can assist with anything else.
If you like, are able to upload an extract from the contract. The definition of tenant in this case is definitely a sticking point which you can dispute.
Have you been in touch with any solicitors?
Thank you for your email.
The wording is definitely ambiguous. I would continue to stand firm on this point as the terms of the agreement should stipulate what will happen in the event of a single tenant leaving.
As this is a proposal, you are free to reject it.