Phase Run's cooldown isn't affected by Watcher's eye's buff

I have Watcher's eye:


the Phase Run skill gem has a 4 seconds cooldown:


The picture of the Phase run is taken with Watcher's eye already inserted into the tree.
Here's a clean one, according to wiki: https://pathofexile.gamepedia.com/Phase_Run

I used stopwatch, and it's still 4 seconds, so not simply a visual bug. I do have Haste aura on.
Another buff from the Watcher's eye, specifically the 32% chaos res, is working fine.

Is there a reason Phase Run isn't affected at all? Do I miss something? Is it not a movement skill? It IS listed as related to movement skill in several tables along wiki, but that's probably not the same as BEING the one.