ilvl isn't necessary everything, and if we look at Heavensward, we can probably predict a bit on how long the weapons can last.
I'll be using Dragoon for my example:
RelicLonginus Zeta: i135
Whites
Titanium Folk: i139
Adamantrite Trident: i150
The Relic weapon greatly outclasses the white gear in weapon damage, and I believe the additional stats do not make up for the weapon damage difference (I don't remember the exact weights). Of course, a High Quality version may be different, but I don't have the numbers for those. In this case, the Relic Weapon could last someone all the way to 60 if they don't get a weapon from a dungeon, and despite the ilvl increase it would be a downgrade from the relic.
Dungeon Weapons
Grasitha: i136
Halonic Ostiary's Halberd: i142
The i136 dungeon weapon and the relic are more or less on par with one another; the weapon damage is the same and the stat differences seem rather negligible. This means the Relic Weapon is still essentially top dog at level 55. Now the i142 dungeon weapon is where the Relic stops becoming Top Dog; +1 Weapon damage, more strength, and I assume secondaries are in a similar position. The i147 requires level 57, meaning your relic weapon is lasting you until you reach 57, and this is assuming the dungeon weapon even drops for you to begin with.
The Relic Weapons found their match at level 55 and their superior at 57 in dungeon drops. This isn't to say that the Anima Weapon will definitely find their match at 65 and their superior at 67, it just says there is a bit of a precedence for the Anima Weapon to last until 67, or all the way to 70 if the dungeon weapons don't drop for them.