This is the reason that they are so upset when faithful spouses dare tell others about their adultery.
Such revelations make them look bad.
It is hard to spin committing adultery in a positive light–especially in a Christian community. (Albeit, some somehow manage to do so–yet, I question how “Christian” that community is that buys such lies and spin.)
This priority of appearances ought to be a clue to outsiders:
In my opinion, you are not working with a repentant cheater when they are consumed with questions regarding how people found out or the faithful spouse exposing the truth regarding their infidelity.
A person who is truly broken by their sin is not worried that people will think badly about them. They are sad that they behaved badly! (And they are committed to do better in the future.)