I can't for the life of me figure out how this proposed system is even supposed to work.

* If you apply the weekly cap to Tank and Healer roles, you may have those players switch over to DPS, which is just going to make DF queues worse.

* A player who is only interested in being DPS is not going to suddenly play a Tank or Healer just because of this system. You can't force people to play a role they don't enjoy. Personally, I happen to play a Healer as my main, but I have tried tanking in the past and it is frankly just never going to be my thing. If I hit my cap on Healer, I would go DPS if I had to. This is the opposite of what you wanted.

* Depending how many hours per week they play, or how they spend their time, some people may not even reach the cap by the end of the week. People who play DPS will naturally reach their cap the slowest, as their dungeon queues are significantly longer. I'm leveling a Rogue alt right now, and dungeon queues are 75+ minutes long, if I ever get in at all. The worry that I may hit a "cap" on dungeons is completely alien to me.

* There are other things to do in the game besides dungeons: PvP, Hunts, upgrading your Relic weapon, daily beast tribe quests, crafting, etc. Hitting the weekly cap on endgame tomestones does not necessarily mean people will happily just queue for dungeons in DF with a new role. They will more likely just play other content and play less dungeons for the rest of the week.

* You assume people who only play DPS even have a Tank or Healer ready for max-level dungeons. But that's not necessarily true. They will have to go back and level one. Even if you implemented this weekly cap tomorrow, it would be a while before we saw any results. But even so, they may go back and simply level another DPS to pass the time, until the weekly reset.