Get value from property on generic class using open definition

Is there a way to get the value of a property from an open type using reflection?

class Program
{
    static void Main(string[] args)
    {
        var target = new GenericType<string>();
        target.GetMe = "GetThis";
        target.DontCare = "Whatever";

        var prop = typeof(GenericType<>).GetProperty("GetMe");
        var doesntWork = prop.GetValue(target);
    }
}

public class GenericType<T>
{
    public string GetMe { get; set; }
    public T DontCare { get; set; }
}

prop.GetValue(target) throws the following exception:

Late bound operations cannot be performed on types or methods for which ContainsGenericParameters is true.

I know I can just do target.GetType().GetProperty("GetMe").GetValue(target), but I want to know if there's a way to get the value without knowing the type.

The simple solution would be to have a non-generic base class that just contains GetMe, but I can't make that change right now.