I've listened to the video a couple times but can't seem to make out what you are hearing. That said... Does the noise change at all when you work the steering wheel back and forth? One weak point on a lot of STIs is the seals inside the power steering pump. Case in point, mine are on their way out at only 70,000 miles, thus necessitating an o-ring and seal replacement at the very least. Anyways, it can sometimes cause a whining noise. I had a power steering pump go out on my old Ford Ranger back in the day and it sounded like I had a supercharger under the hood.
It was ridiculous.
The noise would vary with engine speed, since the pump is obviously run via an engine belt. And of course, as you worked the wheel back and forth the noise would vary as well.
I'm not sure what your issue is since I can't hear it. Frankly, it'd be difficult to identify even if I could hear it. But anytime someone says "a whine', my mind always goes to power steering.
Sent from my SM-N975U using Tapatalk