The fastest human-capable time is thought to be 4:55.96, if you're able to pull off every trick at the right spot. The fastest tool-assisted speedrun is currently 4:54.03, if you use the same timing methods as the human runs (TAS runs start the clock when the game powers on). Tool-assisted speedruns can make use of glitches that require pixel-perfect accuracy.
Either way, it looks like we're getting very close to the fastest possible run.
A note about the timer in the video-- that's controlled by the streamer, so some human error is involved. After finishing it up, software checks the video frame-by-frame to get the official time. The clock in the video says 4:57.02, even though the official time is 4:56.878.