Unless the Police are asleep, these facts make one or both of the participants Suspect Number One. It is Police 101, the most obvious conclusion is usually the correct one, no matter the demeanor of anyone involved or the spin being put every piece of information.
I can only assume LE's discouraging of private searches is because they are postive a crime was commited and they do not want evidence compromised. This may be at odds with why GR discouraged searches, which would be so evidence could be destroyed by time.
So in your eyes an "affair" is the obvious?
what about rock solid alibis?
what about a passed lie detector test?
what about "other info not divulged" ?
would that change your obvious?
(Remember... I cannot release info, nobody is told about lie detector results, so let's look at that as a "hypothetical")