Test error from async function using async/await

Question

I have an async function that throws an error if its argument is falsy:

public async publish(type: string, payload: any): Promise<any> {
  if (!type) {
    throw new Error(`Type is invalid: ${type}`);
  }

  // ... do things ...

  return useNetworkLayerToPublish(type, payload);
}

Then returns a Promise with the network connection results.

It has to be async because its logic uses await.

Before it was an async function the test case was rather simple:

it("should throw an error is type is undefined", () => {
  const test = () => metrics.publish(undefined, []);

  expect(test).to.throw("Type is invalid: undefined");
});

But now I wonder how could I keep this format, I've tried things like:

// awaiting inside expect
it("should throw an error is type is undefined", async () => {
  const test = () => metricsService.publish(undefined, []);

  expect(await test).to.throw("Type is invalid: undefined");
});


// awaiting inside test
it("should throw an error is type is undefined", async () => {
  const test = async () => await metricsService.publish(undefined, []);

  expect(await test).to.throw("Type is invalid: undefined");
});


// awaiting everywhere
it("should throw an error is type is undefined", async () => {
  const test = async () => await metricsService.publish(undefined, []);

  expect(await test).to.throw("Type is invalid: undefined");
});


// passing the Promise directly
it("should throw an error is type is undefined", () => {
  const test = metricsService.publish(undefined, []);

  expect(test).to.throw("Type is invalid: undefined");
});

But none of them work as I expect. Is this even possible or is it the only way to expect the error inside .catch()?