@Bigups well, the explanation given in the chapter is that it will only serve to further the rumours.
As Ju-In said, the current image of Dan-I is that of someone who backstabbed her friend in order to garner the attention of the heavenly kings, though it does confuse me as to why people haven’t seen her with them as a group of friends (then again, I might’ve missed some stuff since I skipped some chaps cuz I haven’t had much time to read lately TwT). I guess because of the incident, people may have become biased and thought that Dan-I was only a try-hard sticking around the group? Anyways, since the situation kinda pits them against each other in the eyes of public, the heavenly kings publicly supporting and standing up for her may just lead the public to think that “Dan-I succeeded in her scheme of getting closer to the heavenly kings since the kings are trying to cover for her despite what she’s “done””, since supporting Dan-I would be “neglecting” Yeoryong (since she’s the victim) and doing the opposite would just plummet Dan-I’s reputation even more.
Think of it like the numerous scandals that we see on the news and internet. We as bystanders or outsiders don’t know the situation, but because of the information that HAS been given to us, it’s possible that our opinions get swayed despite not actually knowing the full situation. In this case, the public is completely convinced that it was Dan-I based on the “evidence”.
This is honestly all I can get from the situation, and it is a bit confusing, but I guess the end point is that public perception and biases can completely twist a situation.