RE: Honest and genuine belief
"....they can start disciplinary proceedings and if at the end of that process they hold an honest and genuine belief that the employee has been dishonest then they will be entitled to terminate that employment on the grounds of gross misconduct."
is the quote in more context. The implication here, I believe, is that after a disciplinary hearing there is still reason to have an "honest and genuine belief" then that could be used as a basis for sacking your arse.
Accordingly one would assume that the disciplinary hearing would have to have gone very badly for the lying employee for such a belief to remain - remember that any formal disciplinary proceeding will be logged and that the interviewee has the right to have someone present as witness / support.
The implication of the phrase "honest and genuine belief" implies to me that the belief is reasonable, well founded and that sufficient steps have been taken to ascertain the facts.
Look, say you lied about having a degree and your employer begins to suspect this some time down the line. They would ask you for proof, such as your certificate, a letter from your university or possibly some other corroborating data (trustworthy witnesses, say) which should answer the question one way or another. Same with professional qualifications, employment history, etc.