Bravo on being one of the only people whose entry is readable when scaled to printer size without the need to squint or pull out a magnifying glass. Already that helps you in one of the voting categories. While I am at scale, I see several of your call out boxes are misaligned. If you do an update, you absolutely want to fix that. It goes a long way towards looking professional and there are people who will absolutely go into a rage over things like that. Doesn't bother me so badly, but I have learned to look for those sorts of things.
Right away I read the introduction and had thoughts. First was that sentence about the sun was a bit purple. I assume you were trying to capture the sometimes purple prose of the pulp science fantasy era. I feel like it went a bit beyond evocative into the odd. Especially when you start calling the sun amoral. Might want to tone that one down just a hair. Heh. Second though is that I was clearly missing something. Sure enough, when I go to the main landing page I find a cover you didn't have space for. Ah-ha! The missing information is on the cover. At least it's somewhere. Can't judge on it, but when the jam is over that's going to be helpful. Especially since there's nothing about which system is being used inside of the document itself without it!
Right away, we get a small 'dungeon' to enter. Super simple 2 room setup, but evocative. The idea of a half-buried tower where you are entering on the second story instead of the first through a window is nice. Extra valuable that the enemy I face inside makes absolute sense for the setting and situation. It's not some stocked monster. Instead it's just a large animal living it's life who happened to find a useful location to stay. I am pretty sure you meant to say red 'circle' though.
Not sure how I feel about the statement on relics. Obviously useful, but it's more that this is a 1:1 ratio situation where each item is practically made for the sole intent of giving them a bonus against that one creature. As to the rest of these first two pages, I don't play Cairn, but I figure I get it. The warping isn't constant, hence the map, etc. I Like that you can cut through the hedge if you want, but given the warping, it may not work as well as you think. Kind of a fun thing since I know at least one player is going to start hacking at bushes eventually.
Key page has some odd grammatical errors you'll want to fix. For 1, I know a few players who are going to obsess over why the rope is there and what they need to use it on. The weird doors are a nice foreshadowing about this place not always working as expected. Water filling it is a solid enough way to say 'don't come back here' without forcing players. Right away I feel bad for the minotaur. Wondering if there's some way to avoid the fight and ease it's situation. Potential ally?
As I read on, there's a lot of permanent body changes here. Certainly in keeping with the weird stuff out of the era of writing, though I suspect a few players are going to be miffed. Imagine if the entertainer PC ends up losing his ability to express happiness! Bonus points if they're the sort who is a hyper-optimistic always-smiling sort. At least with the shackle situation you can just rip the arm off of the skeleton and bring it with you.
And then the ultimate dick move of realizing they didn't have to sacrifice anything to get out. I'm not saying that's a bad thing, just saying it's rough if they realize. My only complaint is realizing the test was just getting to the exit. Prove your worth, stay the beasts, claim the prize. The prize is just having a way out it seems. All the other stuff is offering up something in exchange for something else. If it is worth it or not will depend on the player in question. Since they don't know what they will get for doing so, it's a real crap shoot.
Last page is relevant statistical stuff for the monsters and objects. It's not a bad little one-off. I think how well it goes over depends a lot on the group. I know at least one player who would absolutely flip finding out they made a big sacrifice only to get one of the items in question. Then again, I know another who'd absolutely love the RP that came from the permanent changes. If I was going to change something (other than fixing grammatical stuff or formatting errors) it would be to maybe find a way to make the monsters feel more thematic to the rest of the setting. Minotaurs suit labyrinths, but aren't particularly 'desert' or 'witch' themed. Ooze doesn't feel on brand either, though the tool applying to it makes sense. Then there's a four armed skeleton at the end. Bowing is nice, but again not particularly thematic. One of the biggest strengths is going to be to keep that thematic momentum up from the first tower.
I would suggest creating unique variations on what is here. Maybe the Minotaur is actually a Minodax (Half human, half Addax) With a picture (if you are expanding) to show how it is similar, but more desert themed. The ooze could be a strange sand-ooze so instead of seeing skeletons, it just looks like a pile of sand until it moves and then you see swirling sand inside it! The skeleton can be more or less the same, but add a description of it wearing a light tan thobe and tagelmust to imply it was once a denizen of the desert you arrived from.
Overall, this feels small but complete. The art is limited, but not offensive. It does it's job, which is important. I didn't feel like I needed to go hunting for information elsewhere outside of that initial moment about the setup. I can't speak to how well the stats work since I don't play Cairn, but if I convert them into the familiar games, nothing seemed game-breaking. If anything, I might feel a bit jaded about the tube. Don't get me wrong, as a player I can think of a few dozen ways to abuse the hell out of that ability. I'm just not sure I want to give up my character's facial expressions for it. So yeah. I could see doing this as a one-shot for an evening with friends as a side activity through a main campaign. The sacrifices lose meaning as a true one-shot though, so it's probably best plugged into an existing game where the players are okay with permanent character changes happening unexpectedly.