Enable Both Microsoft Authentication (OpenID) and Custom Username Password authentication in .Net Web App

Question

I'm implementing both Custom Authentication (for External Users) and Microsoft Authentication (for Internal Users). This is a .net 4.7.2 framework webform project.

I have added a startup.cs to the project.

[assembly: OwinStartup(typeof(AppModelv2_WebApp_OpenIDConnect_DotNet.Startup))]

namespace AppModelv2_WebApp_OpenIDConnect_DotNet
{
  public class Startup
  {  
   string clientId = System.Configuration.ConfigurationManager.AppSettings["ClientId"];
    string redirectUri = System.Configuration.ConfigurationManager.AppSettings["RedirectUri"];
    static string tenant = System.Configuration.ConfigurationManager.AppSettings["Tenant"];

    string authority = String.Format(System.Globalization.CultureInfo.InvariantCulture, System.Configuration.ConfigurationManager.AppSettings["Authority"], tenant);

    /// <summary>
    /// Configure OWIN to use OpenIdConnect 
    /// </summary>
    /// <param name="app"></param>
    public void Configuration(IAppBuilder app)
    {
        // Configure Forms Authentication for custom authentication
        app.SetDefaultSignInAsAuthenticationType(CookieAuthenticationDefaults.AuthenticationType);
        app.UseCookieAuthentication(new CookieAuthenticationOptions
        {
            AuthenticationType = DefaultAuthenticationTypes.ApplicationCookie
        });

        // Configure OpenIdConnectAuthentication for Microsoft login
        app.UseOpenIdConnectAuthentication(new OpenIdConnectAuthenticationOptions
        {
            ClientId = clientId,
            Authority = authority,
            RedirectUri = redirectUri,
            PostLogoutRedirectUri = redirectUri,
            Scope = OpenIdConnectScope.OpenIdProfile,
            ResponseType = OpenIdConnectResponseType.CodeIdToken,
            Notifications = new OpenIdConnectAuthenticationNotifications
            {
                AuthenticationFailed = OnAuthenticationFailed
            }
        });
    }

    /// <summary>
    /// Handle failed authentication requests by redirecting the user to the home page with an error in the query string
    /// </summary>
    /// <param name="context"></param>
    /// <returns></returns>
    private Task OnAuthenticationFailed(AuthenticationFailedNotification<OpenIdConnectMessage, OpenIdConnectAuthenticationOptions> context)
    {
        context.HandleResponse();
        context.Response.Redirect("/?errormessage=" + context.Exception.Message);
        return Task.FromResult(0);
    }
  }
}

Login.aspx.cs

 protected void btnInternal_Click(object sender, EventArgs e)
    {
        HttpContext.Current.GetOwinContext().Authentication.Challenge(
                new AuthenticationProperties { RedirectUri = "/" },
                OpenIdConnectAuthenticationDefaults.AuthenticationType);
    }

    protected void btnExternal_Click(object sender, EventArgs e)
    {
        //Assume validation for username password against db is done

        string customUsername = "externaljohn";

        ClaimsIdentity customIdentity = new ClaimsIdentity(new List<Claim>
        {
            new Claim(ClaimTypes.Name, customUsername),
        }, "Custom");

        HttpContext.Current.GetOwinContext().Authentication.SignIn(customIdentity);

        Response.Redirect("Default.aspx");
    }

Default.aspx.cs

if (HttpContext.Current.User.Identity.IsAuthenticated)
        {
            // Get the username
            string username = HttpContext.Current.User.Identity.Name;

            // Now you can use the 'username' variable as needed
            // For example, display it in a label on the page
            lblUsername.Text = "Welcome, " + username + "!";
        }

This is working fine with Microsoft login. When I click "Internal" button, it will redirect to Microsoft login page where I can enter my office 365 credentials and then it will navigate to Default.aspx page and show username correctly.

However, when I click "External" button, it will redirect to Default.aspx but HttpContext.Current.User.Identity.IsAuthenticated is always false.

I haven't done any changes to web.config except adding the appettings for ClientId, Tenant, authority and redirectUri