You must perform Weapon Skills and Skillchains on monsters that check as Easy Prey or higher.
Weapon Skills performed while under Level Sync count.
You receive 1 point for opening/soloing a weapon skill, 2 points for closing a Lv.1 Skillchain, 3 points for closing a Lv.2 Skillchain and 5 points for closing a level 3 skillchain.
You must accumulate 300 total points.
To check to see if the latent effect has disappeared, simply look at your elemental resistances. If you are not receiving the elemental bonus from the Axe of Trials then you have broken the latent. It is NOT necessary to un-equip/re-equip the weapon.
You will not be able to tell if the latent is broken while under the effect of Level Sync. You will not receive the elemental bonuses while Level Sync is in effect. Therefore, you must be unsynced to check.