WebAPI EnableCors with SupportsCredentials = true not working

Question

I have an MVC site deployed to "mysite.mydomain.com" that authenticates with ADFS and creates an auth cookie:

public partial class Startup
{
    public void ConfigureUserAuth(IAppBuilder app)
    {
        app.SetDefaultSignInAsAuthenticationType(WsFederationAuthenticationDefaults.AuthenticationType);

        app.UseCookieAuthentication(new CookieAuthenticationOptions()
            {
                AuthenticationType = WsFederationAuthenticationDefaults.AuthenticationType
            });

        app.UseWsFederationAuthentication(
            new WsFederationAuthenticationOptions
            {
                MetadataAddress = ConfigurationManager.AppSettings["adsfs.MetadataAddress"],
                Wtrealm = ConfigurationManager.AppSettings["adsfs.Wtrealm"]
            });
    }
}

There is also a WebAPI site deployed to "myapi.mydomain.com" with CORS enabled:

GlobalConfiguration.Configuration.EnableCors(new EnableCorsAttribute("https://mysite.mydomain.com", "*", "*") { SupportsCredentials = true });

The user goes to mysite.mydomain.com. The MVC site is authenticated against ADFS and I see that the auth cookie is not set.

My application is mainly SPA, so from javascript there, AJAX calls myapi.mydomain.com using jQuery, setting the withCredentials parameter to true:

$.ajaxSetup({
    xhrFields: { withCredentials: true }
});

It is assumed that these parameters should send security credentials (cookies) to the API. At runtime, I don’t see cookies being set in the API, and I get error 401 (unaronized) as expected.

localhost ( , , ), , cookie, API, . - , (localhost), "mysite" "myapi".

?