Apologies for tardy response. For quick replies, a huge amount of the community are readily available on our Discord server.
Only thing I can think is that you have a different quest active that also involves the library, at which point completing it would enable you to proceed with the mission. That certainly happens across the game at a number of other points.
Other than that, I can suggest filing a bug report on the official FFXI forums in case it's a programming error.