The deep-locus conjecture for positive braid Bott–Samelson varieties

Let $\beta\in\mathrm{Br}^{+}_{n}$ be a positive braid word, let $\mathrm{BS}(\beta)$ be the associated double Bott–Samelson variety, and let $\mathcal{D}(\mathrm{BS}(\beta))$ denote its deep locus. The deep-locus conjecture. The following conditions are equivalent:

1. $\mathcal{D}(\mathrm{BS}(\beta))=\emptyset$.
2. Any two components of the braid closure of $\beta$ are unlinked.

This is presented as a special case of the no mysterious points conjecture. The corresponding stabilizer-locus criterion is proved, but the equality between the deep and stabilizer loci needed here remains conjectural.