Yes and no. You can do t7 with different people during that reset (on which you beat t6), provided no one else in your group has beaten t7 for the week. People can skip turns, but they can't go back to re-do turns. The party leader can only initiate the instance he/she is about to initiate if they are the farthest into the coil for the week (or equally far into it) out of the group.

If you beat t7 this week, you'll still have to do t6 again next week post-reset. (Unless you skip turns knowingly, having the party leader on t7-9)