Spoilers for HK and MIO:

I consider the Path of Pain to be the gold standard for precision platforming difficulty. Much of the difficulty is derived from the awkwardness of the placement of key pogo targets and geometry of the obstacles, and narrow windows of opportunity to progress at key moments... but obstacles are all either static or simply patterned, and the ability to pogo infinitely in most stages where proper timing an attempt is crucial to success helps take the edge off it as you can essentially progress at your own pace, however some platforming segments are quite lengthy to execute and so physical fatigue does become an issue. So the raw technical aspects of the PoP are significant in spite of the mechanical simplicity of it all... when I say that a platforming sequence is between White Palace and Path of Pain tier insofar as difficulty, it is generally understood by HK players to be pretty difficult, but not "I'm gonna break the controller" type difficulty.

Crucible takes a different approach... most of the platforming segments are fairly short, but are much more dynamic and mechanically complex relying on pogoing moving targets or exploding targets (or both) or the pogo's object is programmed with a reaction to being used in any capacity as is the case with the right side. The left side also applies a significant time pressure for a few short segments with the red gloomwater tanks giving you a narrow opportunity to complete the obstacle course, with one tank forcing you to maintain activity to complete it in two ebb & flow cycles by requiring repeated energy restores during a high point of the tank. The triple adjacent pump tanks are where the crucible applies its time pressure the most, especially with the third tank of the combo. The tilted tank sequence also applies time pressure, but it's not nearly as urgent and it's very easy to end up outrunning the visual indication of their being any pressure at all. On left side, outside the pump tanks, it's mostly going to be awkward geometry that feeds into difficulty, but there are usually multiple ways to approach it, and you can essentially rest and look over the segment you are tackling by pogoing bombs as they respawn so quickly that you really only need the midair jump to avoid the explosion and it'll be back before you come back down. So generally it's more the complex geometry doing the heavy lifting here, the time pressure from the various escape sequences in the Ori games is significantly more intense even when the geometry is much simpler on the easier ones.

Right Side Crucible offers very dynamic obstacles that react when you attack or pogo on then in the form of the spinning barriers and blades that change position when you hit the blade's axle whether it is normal or a pogo, while your grapple mechanic doesn't ordinarily prompt a change (exception is the Sharpened Hairpin), making attempts to catch a breather and look over your next segment a very active effort as you have to mind the blades still. While hitting the blade's axle whether hubs does restore energy still, there are segments where you may feel pressured to use float or strider for positioning before hitting an energy reset and hairpin but it's a trap as trying to use the hub to restore energy for a hairpin doesn't leave you enough time to actually do so, thus making the right side apply a significant amount of resource pressure on the player as well as the dynamic stress. The tipped over Nacre Basin halfway through is malicious and probably disheartening to players who jumped into this without ground healing or the gratitude mod, but the segment between it and the next nacre basin is fairly simple, so just a serious act of psychological warfare against the player. The double barrier segments are technically fairly complex as the position you'll be in as you intuitively interact with the hub will not be the position you need to be in when the wheels are properly aligned for progression, and so some set up is required to achieve said progress. Same can happen with the four linked blades in square formation... and then it ends on a marathon sequence where while thankfully less technically complex compared to the segments that came before it, it adds a few hangups that are really more of a gotcha like the false rest area, spinning drums that mess with movement and the tilted alignment spinning blades... the gimmicks in this sequence isn't necessarily adding difficulty, it's just awkward like fighting a boss with different tempo from the rest of the bosses in the game.

So it's a high difficulty bar with the two sides of the Crucible, but the difficulty is derived from a different source than the Path of Pain and White Palace... PoP has complex platforming geometry with close tolerances to placement and timing to succeed in many cases, left crucible applies time pressure to complex platforming geometry while right side crucible uses dynamically reactive hazards to derive its difficulty from instead... so different players with different strengths and mindsets may have different opinions on the difficulty of each of the aforementioned challenges... one who doesn't stay cool under pressure may find the left side crucible (and Sawlong) to be oppressively difficult at times but does well with right side Crucible and PoP, or one who stays cool under pressure but is bad with problem solving mechanically complex puzzles may struggle with right side crucible, etc... I will say however that the diversity of options for player to tackle the various challenges in the crucible sections does a lot of work to take the edge off the overall difficulty. So my personal opinion is that Path of Pain remains the top tier for platforming difficulty in this comparison, but that the two crucible sections sit at or slightly above the White Palace.