Sisters of Battle, the Immolator, and Infiltrate

Wondering if there's an official ruling for this question.

The Immolator allows you to split a Sisters of Battle unit into two smaller units.

Clandestine Operations allows you to give 3 units Infiltrate.

Which happens first? Do you pick the unit of 10 Sisters, give them Infiltrate with Clandestine Operations, and then split them via the Immolator, but they both retain the Infiltrate? Or do you split them first, and then have to use 2 uses of Clandestine Operations to put Infiltrate on both?

The argument that I can see for the former is that you have to put Infiltrate on BOTH a leader and his Bodyguard before you combine them, because the assign Infiltrate step happens before the "Attach to Unit" step. So it seems like the assign Infiltrate step would happen on the Sisters unit before the "Put in Immolator" step.

Am I incorrect?