Should IEquatable, IComparable be implemented on non-sealed classes?

I’ve been thinking about this question for a bit and after a bit of consideration I agree that implementing IEquatable<T> and IComparable<T> should only be done on sealed types.

I went back and forth for a bit but then I thought of the following test. Under what circumstances should the following ever return false? IMHO, 2 objects are either equal or they are not.

public void EqualitySanityCheck<T>(T left, T right) where T : IEquatable<T> {
  var equals1 = left.Equals(right);
  var equals2 = ((IEqutable<T>)left).Equals(right);
  Assert.AreEqual(equals1,equals2);
}

The result of IEquatable<T> on a given object should have the same behavior as Object.Equals assuming the comparer is of the equivalent type. Implementing IEquatable<T> twice in an object hierarchy allows for, and implies, there are 2 different ways of expressing equality in your system. It’s easy to contrive any number of scenarios where IEquatable<T> and Object.Equals would differ since there are multiple IEquatable<T> implementations but only a single Object.Equals. Hence the above would fail and create a bit of confusion in your code.

Some people may argue that implementing IEquatable<T> at a higher point in the object hierarchy is valid because you want to compare a subset of the objects properties. In that case you should favor an IEqualityComparer<T> which is specifically designed to compare those properties.