Does Shadow Strike's flat damage apply twice when dual-wielding?

Shadow Strike includes flat piercing and cold damage in its skill description. I know that flat damage from other sources, like Dual Blades, applies with every application of weapon%. Does the flat damage specifically from Shadow Strike apply to a target twice if I dual-wield?

(I noticed that a 2h weapon was doing less Shadow Strike damage to target dummies than a pair of weapons, despite what the DPS screen claimed, and I want to know if double application of flat damage is why)

Whatever you see on the shadow strike tooltip is per hand, each hand always hits (dual wield)

Nice! Thanks.