Websites Can Identify If You’re Using iPhone’s New ‘Lockdown’ Mode
vice.com
vice.com
This isn't exactly the same, but it's another example of "trying to hide information is, in itself, giving away information".
BTW, tangentially related: I remember reading that the original iPod shuffle function was truly uniformly shuffled, but people started thinking it wasn't and complaining because they'd notice when multiple songs by the same artist played in sequence. So Apple had to make the shuffling less random to avoid those strings so people would perceive it as being truly random.