This is a list of times that I believe could be theoretically possible, but very difficult to get. I'm calling these "bounties" but am not offering any reward. Just consider them to be on the cutting edge of this game.

Classic Mode

1-6 in 6.3s or 6.4s

To achieve this time, you would need to hit the "boulder sky jump" strat. The strat is documented in the run notes of the current IL record. The time it takes for the boulder to roll down the hill is random. I think you'd want a slow boulder to hit a high enough sky jump, but I am not sure. I'm not even sure if it is possible to get a high enough sky jump to make it all the way to the water without hitting land, but I am optimistic that it is possible since the height of the sky jump seems to vary every time I hit it. In any case, this one requires good RNG as well as very precise timing and positioning

4-3 in 9.6s

The downside of using the button control scheme is that everything is either full left or full right. Ideally, going through the gap at the end of the level would barely clear the snow. But I see no other way to do a micro-adjustment in the air that would achieve the desired result other than by using the virtual joystick control scheme. Check out the current IL record.

Piñata Mode

1-6 in 6.3s or 6.4s

If it is possible to get the boulder sky jump strat in Classic Mode, then it should be possible to get the same strat in the Piñata version of the level. Not sure if you'd necessarily clear the blue candy with the same sky jump, but in theory this should be possible.

2-1 in 12.2s

This level is deceptively difficult, as the optimal route across the spring pads is very precise. I think there is room for improvement at the end of the current IL record

3-16 in 17.0s

The current IL record doesn't get the first frictionless landing. Adding this to the run would potentially lower the time by 0.1s. This level is on this list because getting the same rock warp is a shot in the dark, especially after getting on a record-paced run.

Got 3-18 in 6.0s after only a few minutes of grinding. Removed it from the list

Got 2-14 in 15.4s. Playing the level more made me realize that 15.2 might be possible with perfect play. Updated the list

Got 4-8 with a different strat than I anticipated. Removed it from the list

Got 2-14, removed from list

I don’t think 2-5 in 8.9s or 2-18 in 7.5s is possible, removing from list

Got 3-8 in 16.6s. Removed from list

Achieved 2-17 Piñata in 18.1s, removing from list

Got 2-1 Piñata in 12.3s. Updated goal to 12.2

Got 1-21 Classic in 7.5s. Removed from list