Condition lengthening items that don't cause the condition

Does a condition lengthing prefix on a weapon only lengthen conditions when the weapon inflicts the condition on a foe, or anytime? For example:

1) Necro poisons foe. Warrior hits same foe with a Poisoner's Ax. Does necro's poison duration last longer?

2) Mesmer is wielding, but not attacking with, a Cruel Sword (lengthens Deep Wound duration). Mesmer casts Phantom Pain on opponent (which causes a deep wound when it ends). Does the deep wound last longer?

to my knowledge it would have to be inflicted by said weapon. of course i could be completely wrong, who knows