User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-11227630-20170803025655/@comment-5529393-20170803145033

I don't think that it actually is necessary; the inequality is a carryover from Rathjen's notation in his analysis of KP + Pi-3 reflection. I'm guessing it is there to make the OCF more well-behaved.