How can I make a JSON POST request with LWP?

If you try to login at https://orbit.theplanet.com/Login.aspx?url=/Default.aspx (use any username/password combination), you can see that the login credentials are sent as a non-traditional set of POST data: just a lonesome JSON string and no normal key=value pair.

Specifically, instead of:

username=foo&password=bar

or even something like:

json={"username":"foo","password":"bar"}

There's simply:

{"username":"foo","password":"bar"}

Is it possible to perform such a request with LWP or an alternative module? I am prepared to do so with IO::Socket but would prefer something more high-level if available.