I believe they do, but I'm not sure why...
Actually I have to disagree with you unless they've recently changed this. Captured characters keep their counters until rescued, discarded, or removed, however captured characters do lose weapon-class enhancements.
Captured characters with
any placed cards have them discarded when captured by the new game rule, weapons or no.
"Counters", however, are permanent + 1/ + 1 increases. They are not actually tokens, just represented by them. Just like all permanent increases or decreases, they stay no matter where the card goes unless it resets by hitting hand, deck, or discard.
Until a card hits hand, deck, or discard, permanent increases or decreases (including experience credit) stay, regardless of what else happens to the card and are added together to resolve the current */* of the card.
