### 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
int swap_cancel_ok = EDITOR_GET("interface/editor/accept_dialog_cancel_ok_buttons");
	if (swap_cancel_ok != 0) { 
		
		AcceptDialog::set_swap_cancel_ok(swap_cancel_ok == 2);
	}
	ResourceLoader::set_abort_on_missing_resources(false);
	ResourceLoader::set_error_notify_func(&EditorNode::add_io_error);
	ResourceLoader::set_dependency_error_notify_func(&EditorNode::_dependency_error_report);
	SceneState::set_instantiation_warning_notify_func([](const String &p_warning) {
		add_io_warning(p_warning);
		callable_mp(EditorInterface::get_singleton(), &EditorInterface::mark_scene_as_unsaved).call_deferred();
	});
	{
		
		Ref<ResourceImporterTexture> import_texture = memnew(ResourceImporterTexture(true));
		ResourceFormatImporter::get_singleton()->add_importer(import_texture);
		Ref<ResourceImporterLayeredTexture> import_cubemap;
		import_cubemap.instantiate();
		import_cubemap->set_mode(ResourceImporterLayeredTexture::MODE_CUBEMAP);
		ResourceFormatImporter::get_singleton()->add_importer(import_cubemap);
		Ref<ResourceImporterLayeredTexture> import_array;
		import_array.instantiate();
		import_array->set_mode(ResourceImporterLayeredTexture::MODE_2D_ARRAY);
		ResourceFormatImporter::get_singleton()->add_importer(import_array);
		Ref<ResourceImporterLayeredTexture> import_cubemap_array;
		import_cubemap_array.instantiate();
		import_cubemap_array->set_mode(ResourceImporterLayeredTexture::MODE_CUBEMAP_ARRAY);
		ResourceFormatImporter::get_singleton()->add_importer(import_cubemap_array);
		Ref<ResourceImporterLayeredTexture> import_3d = memnew(ResourceImporterLayeredTexture(true));
		import_3d->set_mode(ResourceImporterLayeredTexture::MODE_3D);