Extraordinary claims require extraordinary evidence
I don't walk around recording every conversation I have.
I'd also argue that calling a tech CEO a liar is far from extraordinary. It'd be extraordinary if I accused him of honesty.
Only problem is that the original quote you commented on was not kagi lying, as they actually say the same here
https://blog.kagi.com/waiting-dawn-search
You said that you were not surprised that kagi was lying, only that they were not in this occasion. When you accuse somebody of lying it makes sense to provide at least some evidence of that.
At the very least, they are very clear about which indexes they use and how.