The previous special round I killed the last dog then activated a TEDD task. It started the task, time changed, then the task disappeared. A few rounds later I went to check the temporal current and it said it was still recharging. This is the next special round I had. You can see the time never skips forward, and the rift is still "recharging."
This could be useful if you're trying to do the pile up glitch in the house. I believe this can be recreated by taking a TEDD task after you kill the last dog.
ETA: I specifically got the melee kills TEDD task, and I did the drinking animation for the Melee Macchiato perk after the time skip, and it never added the badge or gave me the perk, only did the drinking animation.