A Conditional Acceptance is used when a demand for payment or performance (a ‘presentment’) is made upon you to which you do not agree. You accept the adversary’s presentment and agree to perform as they demand but only upon receipt of their proof of claim that you have any liability to them, such as a lawful contract or any other proofs you may need to convince you of the validity of their demands.. This is a way to avoid arguing so that should there be an attempt at litigation, it can be shown that there is no controversy since you never refused to perform but merely asked for some proof of why performance on your part would be appropriate. Cases are brought to court for argument. If there is no argument, there is no case.
Your documents are most effective when served by a notary public with responses directed to the notary so that at the completion of the process, if the matter is not settled, the notary can issue a detailed Notice of Non-Response.