### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
Ref<Environment> fallback = get_tree()->get_root()->get_world_3d()->get_fallback_environment();
	if (fallback.is_valid() && fallback->get_path().is_resource_file()) {
		HashMap<Ref<Resource>, bool> processed;
		_find_and_save_edited_subresources(fallback.ptr(), processed, 0);
		save_resource_in_path(fallback, fallback->get_path());
	}
}
void EditorNode::hide_unused_editors(const Object *p_editing_owner) {
	if (p_editing_owner) {
		const ObjectID id = p_editing_owner->get_instance_id();
		for (EditorPlugin *plugin : active_plugins[id]) {
			if (plugin->can_auto_hide()) {
				_plugin_over_edit(plugin, nullptr);
			} else {
				_plugin_over_self_own(plugin);
			}
		}
		active_plugins.erase(id);
	} else {
		
		
		List<ObjectID> to_remove;
		for (KeyValue<ObjectID, HashSet<EditorPlugin *>> &kv : active_plugins) {
			const Object *context = ObjectDB::get_instance(kv.key);
			if (context) {