The optional press can only be used during the current round.
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Mighty Grapple
Grapple.
Press, only usable to continue combat.
Strike: hand strike or use a melee weapon strike. This strike is at +1 damage, with an optional press only usable to continue combat.
Rulings
Adding damage to a strike that does not deal damage has no effect, so the strike still will not do damage.
Additional damage inherits all of the properties of the base damage.