The ground work has been made in terms of updateVueRouteOnSearch to accept an object with the property pushHistory set to true. Currently it will do the switch to push to history, but searches are not currently triggered when going back and forth through history.
We need to figure out how that exactly should be done, as this will both have to update the internal state, but also possibly update fields to reflect the changed state, which may affect other components. It needs a bit of thinking through.