Does ambidexterity have any effect while equipping a bow?

Ambidexterity is supposed to grant increased damage on main hand and increased speed on off hand.
Doubt is, if your character is wielding either a two-handed melee weapon OR a bow and a quiver will it still provide at least a partial effect?

Two-Hand Damage is Main-Hand Damage, so yes, technically, that node will work... but none of the small nodes leading up to it will work. Bow Damage can never be Dual-Wielding Damage; quivers aren't technically weapons.

So, that means a two-handed mace will get the dmg increase from the passive, but not the extra speed?
And would a bow behave in the same way?

Thank you!