Volcanic Range is not one of the oldest dungeons on record, but in the years it's been around for it's gotten quite a fearsome reputation—and, really, the fearsomeness to back it, too. It's one of the somewhat rare dungeons that doesn't even try to kidnap people at all, opting instead to send out monsters to find people and bake them alive. And of course, people under these monsters' ministrations take a lot longer to die than they would if they were exposed by these amounts of heat from any other sources, as dungeons always try to keep their victims alive for as long as possible. For this dungeon in particular, the survival rates of people who escape the monsters is abysmal, since once outside their zone of influence the tissue damage will often be sufficient to be lethal, so it is paramount to not let monsters escape at all.
The inside of the dungeon is open air: there is a sky, and an expansive horizon. But the place is blazingly hot, decorated by crisscrossing rivers of lava, burning forests, and active volcanoes. The sky is perpetually a deep dark red that reflects the fires, and burnt and burning ashes fly in the scorching wind. What's more, the dungeon is old enough that it has in fact expanded to cover a lot of its horizon, and at least some of the forests and volcanoes visible in the horizon are actually real and visitable. The terrain is rocky, dry, and uneven, with the mirages helping it conceal numerous tripping hazards lying in wait to get you to fall on your face and suffer a third-degree burn or worse.
If that were all it was, though, it would not be an S-rank, even with some reasonably scary monsters.
First, it kills comms and electronics. They're all advised to just leave phones and other gadgets behind, because they just won't be useful in there. They'll need to navigate using their regular senses, no tech assistance available. Not just that, but sensors have a range limit enforced upon them, when they're not entirely noped by the dungeon, so they can't even compile a very good map of the dungeon in any way other than literally going out and seeing.
Second, you can't stay anywhere for too long, as the rocks that make up the ground will literally start animating itself into golems anywhere there's anyone, and the place's got lava just below the surface everywhere (and if that's not realistic: fuck you, the dungeon does not care about realism, it wants you to burn) so that continually destabilises the terrain. That also means that they can't do the normal thing of having a medical team stationed inside right next to the portal that they can do with other dungeons as they would probably all be killed and the entrance area would be made impassable.
Third, the place is huge. That's it, that's the third item, it's just had so long to grow and searching it for the core is really difficult.
Fourth, there are dragons, some larger than mammoths, all with scales tougher than steel and capable of breathing fire that melts rock.
Fifth, every monster it produces explodes on death, Balrog style. The dragons' explosions are enormous, while the golems' get larger depending on how long it's been since they spawned (so you want to kill them quickly) (and so, of course, if a golem does break past the dungeon portal that is an emergency as killing it in an area with civilians will cause unacceptable levels of collateral damage.
And finally, sixth, like many other S-rank dungeons it produces numerous identical decoy "cores" that all monsters will protect with their lives and which, if destroyed don't do anything. ...well, besides exploding.
In summary, no one actually has any ideas for how to permanently close this dungeon. Whenever it appears they try to systematically explore it, get attacked by monsters, occasionally find a core-alike, draw a specialist sensor in to verify whether it's the real core, it never is, they try destroying it anyway, it explodes, the dungeon remains. So the best they can do, really, is keep the monsters at bay until the dungeon decides to leave. Typically, this is done by sending a team in, staying in for two to six hours, and then being replaced by a new team later, rinse and repeat.
This is not to say, of course, that they're not going to be trying. Two of the five pairs—Tae-gun and Hye-jin being one, Rodriguez and her partner Montenegro being the other—have long-range exploration as their duty. They'll be going in opposite directions and trying to find something, anything that might prove to be the real core. They're chosen for their extreme mobility, of course, with Rodriguez being able to fly herself and her partner using her telekinesis and Tae-gun being able to do the same with his ice constructs, but now that Tae-gun has a partner who can help with his backlash he is also a damage-dealing powerhouse himself, which will allow him a lot more flexibility in his job.
High damage-dealing, mobility, and defensive capabilities: you need at least two out of those three to be of meaningful use, here. Tae-gun, with Hye-jin, now has all three. Hye-jin has two.