onEmbeddedMessagingConversationOpened event can be listened to detect or check whether page occurred for the Salesforce Messaging for Web.

Event Listener Code:

You can use the following code to listen to the onEmbeddedMessagingConversationOpened event for your reference.

/* START:: Conversation Opened Listener */
window.addEventListener( "onEmbeddedMessagingConversationOpened", ( event ) => {

	console.log( "START:: Conversation Opened" );
	console.log( "Page location is " + window.location.href );
	console.log( "Event detail: ", JSON.stringify( event.detail ) );
	console.log( "END:: Conversation Opened" );

} );
/* END:: Conversation Opened Listener */

You should add the above code along with the Embedded Service Deployment code snippet.

Salesforce Article:

https://developer.salesforce.com/docs/service/messaging-web/guide/event-listeners.html