Reimian Empire

If that's the case, there's no point in doing the quest anymore, even if it's only a couple of days ahead.

Well, if that's true, then there's no reason to hold up the others.
Bedivere successfully negotiated with the captain of the guard. Over the next couple of days, he and Tsukiko visited all possible witnesses, checking and rechecking their stories. At some point, the information reached the two that one of the suspects, Elias, had suddenly died. The city guard went to the baron's house to investigate, and the assignment at "Morpheus" was canceled.
Yes, I added a lot of details, but since you say the quest is pointless, it seems that everything was found out without us after the baron's death.