OWIN open id - Authentication.SignOut() not working properly

Question

Background

I'm newbie in azure, and this is my first post so please bear in mind with me.

Currently I'm working on Azure Single-Sign-On for my company.

Following the instructions: Web App Sign In & Sign Out with Azure AD, I've created/copied a MVC 5 project with OWIN OpenId Authentication using VS2015.

Issue

Signing in is working well but signing out is not:

1. signed in with user A and signed out. No problem.

2. And then tried signing in with another user B: clicking 'Use anther account' in login.microsoftonline.com and entering user B's email address, without allowing me to enter the password, it redirected me to my home page showing user A is signed in!

Only if I closed and reopened the browser (tried with both Chrome and FF), it allowed me to enter user B's password and to sign in as user B.

Codes

public class AccountController : Controller
{
    public void SignIn()
    {
        // Send an OpenID Connect sign-in request.
        if (!Request.IsAuthenticated)
        {
            HttpContext.GetOwinContext().Authentication.Challenge(new AuthenticationProperties {RedirectUri = "/"},
                OpenIdConnectAuthenticationDefaults.AuthenticationType);
        }
    }

    public void SignOut()
    {
        string callbackUrl = Url.Action("SignOutCallback", "Account", routeValues: null,
            protocol: Request.Url.Scheme);

        Request.GetOwinContext().Authentication.SignOut(
            new AuthenticationProperties {RedirectUri = callbackUrl},
            OpenIdConnectAuthenticationDefaults.AuthenticationType,
            CookieAuthenticationDefaults.AuthenticationType);
    }

    public ActionResult SignOutCallback()
    {
        if (Request.IsAuthenticated)
        {
            // Redirect to home page if the user is authenticated.
            return RedirectToAction("Index", "Home");
        }

        return View();
    }
}

I did a lot of searches but still no luck.

Here is the exact same issue as mine: OWIN - Authentication.SignOut() doesn't remove cookies

What @user3613871 said makes a lot of senses to me:

the cookies are AppServiceAuthSession cookies - no longer the AspNet cookies. Thus, the logout no longer works.

But I still couldn't figure out the workaround.

Is there any thing I can do on the settings in azure or my codes to clear up "the azure cookies"?

Any idea is welcome and thanks in advance.