Honestly, even if you're hell bent on the idea of timed runs, there are better ways to do it. I understand that some hardcore players currently don't believe content can be made difficult or entertaining without a timer. The reality is though, for those who feel this way, the current content isn't difficult, it's just giving you a score to beat. How many seconds can you shave off?

If you're going to win with 5 chests one way or another, the amount of seconds you can shave off is almost entirely irrelevant.

What having the entire dungeon be one timed segment does, is artificially extend the challenge. Trying to get a good strategy down for the last boss? You gotta run the entire instance as many times as you want to try the boss. This isn't challenging, if you've already mastered the rest of the dungeon, it's just a giant time sink.

If time trials have to be here (which honestly, I still don't think they do) let's do segmented time trials, with resting areas in between. Think those old rooms in FF dungeons with save points, where you could set up a tent and rest your party. After completing a segment, you can rest, then go for the next segment when your group is ready. Wiping means you start the segment over, but your clear time for the previous segment is stored. You could even allow players to re-attempt individual segments. (And obviously time in the rest areas won't count towards the total timer.)