[Discover] "Unsaved changes" badge should appear after updating the time picker for a saved search with "Store time with saved search" setting

[jughosta]

Kibana version: 8.12

Describe the bug:
Updates to the time picker should trigger "Unsaved changes" badge in Discover if the saved search has "Store time with saved search" enabled.

Steps to reproduce:

1. Save a new search with "Store time with saved search" switch on
2. Change the time range in UI or change the refresh interval setting
3. Notice that "Unsaved changes" badge does not appear.

If user applies other changes which do trigger the badge, then pressing "Revert changes" restores the saved search correctly including the time filter settings. The problem is only that the badge does not appear right away.

[elasticmachine]

Pinging @elastic/kibana-data-discovery (Team:DataDiscovery)

[kertal]

Some remarks, we currently don't update saved search when just the time range changes, this could be done like this

Adding

   const timeFilterSubscribe = services.timefilter.getFetch$().subscribe((next) => {
      if (savedSearchContainer.getState().timeRestore) {
        savedSearchContainer.update({});
      }
    });

After this code:

https://github.com/elastic/kibana/blob/cb19e99b382d1a8a55a46e53b21fe15eee513d3a/src/plugins/discover/public/application/main/services/discover_state.ts#L391-L390

However, reverting the change with the Unsaved changes badge didn't work on the first click, but on the second, so there's high likely more to do in this case