Anti-diagonal core conjecture
Determine whether, for the reduced-word BFZ deep locus of the largest open double Bruhat cell G^{w_0,w_0}\subset SL_N, the anti-diagonal torus A_N^\circ is contained in every irreducible component, or at least every component invariant under anti-diagonal transpose, and prove whether A_N^\circ is exactly the universal intersection core of the irreducible components containing it.
References
This proposition leaves open another question: is the anti-diagonal torus itself an irreducible component, or is it always contained in larger deep components? The $SL_3$ calculation suggests the latter: the anti-diagonal support stratum is contained in several larger components, and we conjecture that in higher rank, the anti-diagonal may be better understood as a universal intersection core of the largest-cell deep locus.
For $G=SL_N$, the anti-diagonal torus $A_N\circ$ is contained in every irreducible components of $\mathcal D{w_0,w_0}$, or the ones that are invariant under anti-diagonal transpose.
More generally, $A_N\circ$ should be exactly a universal intersection core, which means its is the intersection of all irreducible components containing it.