### 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<ResourceImporterScene> import_animation = memnew(ResourceImporterScene(true, true));
		ResourceFormatImporter::get_singleton()->add_importer(import_animation);
		{
			Ref<EditorSceneFormatImporterCollada> import_collada;
			import_collada.instantiate();
			ResourceImporterScene::add_scene_importer(import_collada);
			Ref<EditorOBJImporter> import_obj2;
			import_obj2.instantiate();
			ResourceImporterScene::add_scene_importer(import_obj2);
			Ref<EditorSceneFormatImporterESCN> import_escn;
			import_escn.instantiate();
			ResourceImporterScene::add_scene_importer(import_escn);
		}
		Ref<ResourceImporterBitMap> import_bitmap;
		import_bitmap.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_bitmap);
	}
	{
		Ref<EditorInspectorDefaultPlugin> eidp;
		eidp.instantiate();
		EditorInspector::add_inspector_plugin(eidp);
		Ref<EditorInspectorRootMotionPlugin> rmp;
		rmp.instantiate();
		EditorInspector::add_inspector_plugin(rmp);
		Ref<EditorInspectorVisualShaderModePlugin> smp;
		smp.instantiate();
		EditorInspector::add_inspector_plugin(smp);
	}
	editor_selection = memnew(EditorSelection);
	EditorFileSystem *efs = memnew(EditorFileSystem);