Speculation incoming, but I am pretty sure they will mostly go to waste.
We don't know exactly what will happen to our relic in the expansion, however the devs have already stated there will NOT be a zodiac weapon for the three new classes. With that information alone we can near certainly say the new weapon quests, whatever they may be, will not be excluding the three new classes, therefore will not be using the current relic as a required base weapon. Nor do I believe they will give any significant advantage due to the fact that it would be quite dissuasive to attempt the new jobs.