There is no Underdeep offline mode: the game checks the network and the system clock when it starts and shuts down if that check fails, and nothing at all accrues while the game is closed. There is no offline progression and no background catch-up.
Underdeep offline mode: there is none
The demo behaves the same way: it refuses to launch if it is not granted internet access. The store page lists a broadband Internet connection in both the minimum and the recommended requirements for Windows and Linux, so the requirement is documented rather than accidental.
The startup check that stops the game
The game performs a network and system-clock check when it starts and shuts down if that check fails. Those checks are what the "bad connection" and "clock is wrong" messages on the startup screen mean.
If the game refuses to start, the useful checks are: make sure the system clock is correct, verify the game's files through your launcher, and try a different connection. Players in some regions report that the launch failure tracks their provider or their location, and that switching routes fixes it. The full recovery procedure, including the save migration, is in the known problems guide.
Nothing accrues while the game is closed
The idle loop runs on loop tickets and the free cycle timer while the game is open. Nothing accrues while the game is closed, and there is no background catch-up, which is why a full day of tickets is worth about fifteen hours of idle play and the other hours simply do not exist. What those tickets do is covered in the loop ticket guide.
Why the game needs a connection
The reason given for the requirement is the asynchronous multiplayer: the scoreboard, the friend system and the gift exchanges all resolve against the server, and your character can meet another player's character inside the dungeon. That feature is described in the other players guide.