What happens to an escrow holdback if neither side agrees a claim against it is valid?

The disputed funds generally stay put with the escrow agent, often a law firm or trust company, until the disagreement is resolved through whatever mechanism the purchase agreement specifies, rather than being released to either side based on one party's own view of who is right. Escrow agents are typically neutral parties who release funds only on joint written direction from both the buyer and seller, or on receiving a court order or arbitral award directing them to do so.

How the underlying disagreement itself gets resolved depends on the agreement's dispute-resolution clause: purely financial or accounting disputes are sometimes referred to an independent accountant for a binding determination, while broader disagreements about whether a representation was actually breached, or whether a claim is otherwise valid, more commonly go to negotiation, arbitration, or litigation as the agreement specifies. Either party wanting the disputed funds released should expect this to take real time and, likely, formal process, rather than assuming a stalemate simply resolves itself.