Rational smoothness and pattern avoidance for arbitrary involutions

This is in effect part 2 of a talk I gave in this seminar in 2012 (!)  I will prove a conjecture that I made in that talk, characterizing which orbits of the even complex orthogonal group \$O_{2n}\$ in the flag variety for \$GL_{2n}\$ have rationally smooth closure via a pattern avoidance criterion.  The odd case remains tantalizingly open, though I conjecture that the same list of bad patterns characterizes rational smoothness of orbit closures.