### 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
dependency_errors.clear();
	Error err;
	Ref<Resource> res;
	if (ResourceLoader::exists(p_resource, "")) {
		res = ResourceLoader::load(p_resource, "", ResourceFormatLoader::CACHE_MODE_REUSE, &err);
	} else if (textfile_extensions.has(p_resource.get_extension())) {
		res = ScriptEditor::get_singleton()->open_file(p_resource);
	}
	ERR_FAIL_COND_V(!res.is_valid(), ERR_CANT_OPEN);
	if (!p_ignore_broken_deps && dependency_errors.has(p_resource)) {
		Vector<String> errors;
		for (const String &E : dependency_errors[p_resource]) {
			errors.push_back(E);
		}
		dependency_error->show(DependencyErrorDialog::MODE_RESOURCE, p_resource, errors);
		dependency_errors.erase(p_resource);
		return ERR_FILE_MISSING_DEPENDENCIES;
	}
	InspectorDock::get_singleton()->edit_resource(res);
	return OK;
}
void EditorNode::edit_node(Node *p_node) {
	push_item(p_node);
}
void EditorNode::edit_resource(const Ref<Resource> &p_resource) {