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Unlike the Catman achievement, you can unlock this one after finishing the game simply by reloading the pavilion levels.
I recommend using a stopwatch to check your progress, as the game provides no internal time indicator.
Another important thing to be aware of is that the counter seems to be bugged. After finishing several plays with a stopwatch, and comparing the result with the counter value in the save file, I noticed that the game adds about 20 seconds to the time I get with the stopwatch. My guess is that the counter starts as soon as the level loads, and NOT when you hear the start signal. Skipping the level intro doesn't seem to help either.
That's why I recommend aiming for a time less than 3 minutes 40 seconds. It should be enough to get this achievement.
Another tip is to store the cubes in your inventory (E) until it's full, and only then throw all the cubes in your inventory into the current. It saves you some time compared to taking and throwing each cube one by one.