Kyle was feeling rather pissed off.
Royally pissed off, actually.
He was well aware of the rumour that was going on.
He was extremely angry that Sayuri was maligned in such a way. She was innocent, with the most beautiful soul ever, and they called her all sorts of slurs. He took note of those that did so.
He really had take his hat off to Sarah.
As much as he hated the rumour, though, there was absolutely nothing he could do about it right now. If he actively stepped in to quash it, it would only cause more harm. By denying the rumour, they'd only think he was trying to cover it up and protect Sayuri. That he was being a 'gentleman' and a 'man of honour'.
Worse, he had no evidence to pin the start of the rumour on Sarah.
If he did, he would have had some valid reason to fire her. After all, the standard employment contract of Smith Industries had one unusual clause.
No gossiping or spreading false information.
