Is there a single word that has the same meaning as "As far as I know"?

You have discarded "apparently" because of the lack of confidence it conveys. I submit that AFAIK makes the finitude of certainty it's central thesis. "Seemingly," or "apparently" connote an understanding to the limit which the senses can detect a thing, and as such, are seemingly the best single word options available. "Presumably" connotes some inferential leap, a process of induction, or information obtained from sources other than oneself (purportedly, ostensibly). These connotations are not mutually exclusive. A thing may be apparently or seemingly true based upon inference or second-hand information. But these adverbs may also be used to connote a direct connection between observer and the observed. A thing is known to the observer as it appears and as it seems. You might counter that one can always distrust his/her own direct sensorial experience as well. But then the question is no longer to find a word for "how far I know a thing," but rather, "what does it mean to know a thing..." and that's a different Stack Exchange altogether.