Finally, an offensive run.

I used https://one.livesplit.org/ and manually started it / stopped it with Alt-Tab. For that reason, the clock starts a bit sooner and ends a bit later. The timer is visible on the video thanks to OBS scene with multiple sources. The audio cuts off between 1:40 and 1:43 in the video because TF2 was not focused. Going through the captured video, I can see that the text "SETUP" disappears at 01:43.11 (11 frames out of 60 fps) and the victory sign appears at 05:50.28. This makes a total run of 4:07.28

My strategy for the speedrun is that blue bots flock at the cart already, so as long as I remove all obstacles from their path, they will push at maximum velocity. The enemy bots have their quirks at detecting spies, but they are generally pretty naive and can be easily killed.

I went through the video and noted down each time section when blue team is not pushing the cart. Adding the times of those 14 sections equals to 82 seconds. In other words: if a super efficient player manages to clear all enemies in front of the blue team, so that the blue bots will constantly push, then the speedrun timer could go down as little as ... 2:45

Additional inefficiencies in my run are time segments where only 1 or 2 players are pushing the cart (less than maximum speed) and I haven't gone into the math of calculating how much time was spent going through the same track section twice due to the cart rolling back at times. If those are taken in consideration, then the theoretical minimum would be closer to 2:35 (I guess)