It was during his time pondering earlier that he realized something.
Since the Emham Forest operation dealt with Prof. K and Giana is exposed and then got attacked by Awakened from 25 Golden Crest Families, Rex is quite sure that it was non the two's fault that causes this.
The feeling he has towards them doesn't hinder the fact that the two were capable.
One is a messenger from the SCO and the other is one of the pillars of humanity, doubting their competence would be foolish of Rex when so much is at stake. But he reckons when Prof. K came earlier, he only said that the operation got exposed.
If it were his fault, then he would've been more apologetic when he report to Rex.
But he was clearly not, he was more concerned than apologetic.
