  High-eccentricity migration is a likely formation mechanism for many observed
hot Jupiters, particularly those with a large misalignment between the stellar
spin axis and orbital angular momentum axis of the planet. In one version of
high-eccentricity migration, an inclined stellar companion excites von
Zeipel-Lidov-Kozai (ZLK) eccentricity oscillations of a cold Jupiter, and tidal
dissipation causes the planet's orbit to shrink and circularize. Throughout
this process, the stellar spin can evolve chaotically, resulting in highly
misaligned hot Jupiters. Previous population studies of this migration
mechanism have assumed that the stellar spin is aligned with the planetary
orbital angular momentum when the companion begins to induce ZLK oscillations.
However, in the presence of a binary companion, the star's obliquity may be
significantly excited during the dissipation of its protoplanetary disk. We
calculate the stellar obliquities produced in the protoplanetary disk phase and
use these to perform an updated population synthesis of ZLK-driven
high-eccentricity migration. We find that the resulting obliquity distribution
of HJ systems is predominantly retrograde with a broad peak near 90$^\circ$.
The distribution we obtain has intriguing similarities to the recently-observed
preponderance of perpendicular planets close to their host stars.
