Should an integrated crown race of an carbon fork be greased before assembly?

It seems like the fork has an integrated crown race. Should it be greased before assembly and pre-loading?

Current headsets have a conical or spherical seat for the cartridge bearings. The purpose of this seat is to absorb shocks that the bearing would be unable to otherwise absorb. This absorption requires the seat to be greased. With no grease, it won't absorb these shocks as reliably as it would with the grease.

Why is this shock absorption so critical, then? Before the conically or spherically seated cartridge bearings, headsets had the bearings directly running on the crown race and the cups. Because these bearings cannot absorb shocks similarly, older bicycle headset bearings tended to develop so-called "indexed steering" where the steering of the bicycle has a notched "home" position in the middle so that turning the fork away from the unsteered position required slight force.

This "indexed steering" is not dangerous but it is annoying.

If you leave the conical/spherical seats of the cartridge bearings ungreased, you are increasing the risk of the headset bearings developing this "indexed steering" annoyance.

Thus, all (top and bottom) interfaces of both headset bearing cartridges require lubrication.

My gut instinct says no grease. Point one, The Park tool manual makes no mention of applying grease to the fork. Second point is that generally you don't want grease in continuous contact with carbon fiber. Third point is that by lubricating the interface you risk the inner race spinning on the fork. The idea is for the fork and inner race to move at the same time, while the outer race stays fixed in the headtube. If the inner race were to slip on the fork it could cause severe damage to the fork. It is the same logic as to why you don't grease handlebar clamps or stems.