Raid Boss "sigil" question

Do sigils carry over to the next raid?

Our guild has finished closing portal 10. There are still guildmates who have sigils left because others have already used up all of theirs to close portal 10

Will those sigils stay in their inventory for use the next time Raid Boss happens in rotation?

More importantly, if others were to keep throwing away gems at buying that last pack, would those sigils stay in the players inventory as well to give a head start on the next raid?

No. If they don’t use them this week they will be lost.

Of course they don’t. It would mean less people would buy them.