I think Loki reported as having killed the king, risk Elbaf getting burnt down, and his brother describing how bad of a guy he is might suffice.
With the exception of commiting arson, aren't those allegations though?
The GK went directly towards Loki in order to recruit him, I think it's because of his devil fruit. Something happened where the fruit almost fell into the wrong hands and Loki took questionable actions in order to protect Elbaf's Legacy and throne. He ate the fruit. And if he did kill his father, maybe it was an accident or maybe Harald did deserve it. Maybe the World Government killed Harald for the fruit and Loki snatched it before they could, which would explain why they're trying to recruit him so intentionally.
On the subject of murder allegations:
Sabo was reportedly King Cobra's killer, Luffy was reportedly Vegapunks' killer. See the pattern? We need to hear from Loki that he didn't or did kill his father.
Loki is mischievous, like a grown adult with a toxic attention seeking disorder. He's the perfect example of a privileged kid growing up in an abusive relationship with his parents, in this case his father.
From the very beginning, Oda has been very intentional describing parent and child relationships. And lately we've seen extreme cases where the relationships are so toxic that someone ends up dying or in the verge of death.
Dragon and Luffy
Judge and Sanji
Big Mom and her children
Kaido and Yamato
Kuma and Bonney
Garp and Aokiji (sort of)
Cobra and Vivi
Imu and Gorosei
Garling and Shanks/Shamrock
Harald and Loki/Hajrudin
I'm probably missing some, but they are there.
All of that to say that it is too early to come to a conclusion and that instead of judging Loki maybe we should practice good old curiosity.