Professor Nomos said:
well its not that either.
Rather its that words like remove, reduce, lose etc can only change printed icons, str, cost, traits etc. (unless of course printed is in specific card text such as the Maester's path agenda).
Icons that are "gained" such as those granted by the NW agendas can never be removed from a character.
Not true. Loses/Reduce etc applies to gained and printed properties of a card.
On a side note, I don't believe there are cards that say "lose all icons" and only see cards whose effect is losing one of each icon. One of several examples is Orphan of Greenblood:
"Any Phase: Discard Orphan of the Greenblood from play (cannot be saved) to choose 1 character. Until the end of the phase, that character loses a Military, an Intrigue, and a Power icon."
I think the wording is nearly identical to that for the other effects that have you lose 1 of each.