Httpapi not deploying when new stack created when using custom domain, and not accepting base path of '/'

I tried to follow the description of how to setup a base path mapping here:

https://docs.aws.amazon.com/cdk/api/latest/docs/aws-apigatewayv2-readme.html#custom-domain

My actual code was:

const domainName = "test.testdomain.com";

const dn = new apigw.DomainName(this, "DN", {
  domainName,
  certificate: certificate,
});

const httpApi = new apigw.HttpApi(this, "HttpApi", {
  defaultDomainMapping: {
    domainName: dn,
    mappingKey: "hello",
  },
});

When I run this with a stack that has never been deployed before, I get the following error:

Invalid stage identifier specified (Service: AmazonApiGatewayV2; Status Code: 400; Error Code: BadRequestException;

Secondly if I try to change mappingKey to '/' I get the following:

An ApiMapping key may contain only letters, numbers and one of $-_.+!*'(), (Service: AmazonApiGatewayV2; Status Code: 400; Error Code: BadRequestException;

Any help you can give is really appreciated, as this is such a great feature, and when it's possible to get authentication in there too, that will be icing on the cake!

Error Log

Environment

• CLI Version :
• Framework Version:
• Node.js Version:
• OS :
• Language (Version):

Other

This is 🐛 Bug Report