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I think it very unlikely that BAM has been taken offworld. For one thing, when you get hold of something this dangerous (and being able to start a desolation without an actual desolation falls into that category) you tend to keep it close. I don't think the Elsecallers would have just given the gem out of hand and - to be honest - as sprens are native to Roshar and Surgebinders don't seem to be ageless it would be seemingly impossible to maintain an Elsecaller exclave somewhere offworld (except maybe on .Ashyn, idk if there are spren there, do we have a WOB on that?) Another thing left unanswered would be Gavilar's access to Voidlight. We have from the Queen that Gavilar had discovered an Unmade but never learned to bond it. Since Yelig-nar doesn't seem to be very secretive with his method, I think it must have been another Unmade. Keeping with the pattern that the other Unmade stayed in hiding until the Desolation started it seems most likely to be BAM or (keeping in mind his very uncertain true motives) Sja-anat, though i think BAM a lot more likely considering the surrounding lore being filled with references to her providing Voidlight and the ability for parsh to bond with Voidspren, an ability Gavilar almost outright promises the Parshendi. Edit/Note (in case that wasn't considered canon already): https://wob.coppermind.net/events/175/#e8400 basically confirms that MAB was captured during the False Desolation