Preventing ASP.NET Identity from creating its default cookie when specifying different cookie scheme

[mshwf]

I have two separate groups of users. I want to provide different login/logout screens for each group.
the way I figured that out is by using something like this:

if (group1) // default
	Challenge(); 
else if (group2)
	Challenge("SecondScheme");

In the Program:
I have:

builder.Services.AddIdentity<ApplicationUser, IdentityRole>()
    .AddEntityFrameworkStores<ApplicationDbContext>()
    .AddDefaultTokenProviders();

To support this scenario I added a cookie:

builder.Services.AddAuthentication()
.AddCookie("SecondScheme", options =>
{
    options.LoginPath = "/SecondAccount/Login";
    options.LogoutPath = "/SecondAccount/Logout";
});

But whenever I login with the second scheme, the .AspNetCore.Identity.Application cookie is created storing the user principal, along with the second scheme's cookie, despite I'm specifying the second scheme when I log in:

 var claimsIdentity = new ClaimsIdentity(
 [
     new(ClaimTypes.Name, SelectedAccount.UserName),
     new(ClaimTypes.NameIdentifier, user.Id)
 ], "SecondScheme");
await HttpContext.SignInAsync("SecondScheme", new ClaimsPrincipal(claimsIdentity));