The SigningCoordinator facilitates the signing process between the service worker and the UI script.
When a Witnesser (managed by an ObservableWallet in the service worker) issues a signing request, the SigningCoordinator (acting as a proxy) issues that request in the UI script context. This allows the extension to handle user interactions, such as hardware wallet communication or manual transaction confirmation. Once signing is complete, the SigningCoordinator sends the signatures back to the Witnesser in the service worker.