Yes. Any equipment pre HoW will require ascendant/radiant materials. In order to exchange materials at the speaker you gotta have them in your character inventory.
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What about armor?
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Yep, armor too.
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Oh okay, thanks you guys! I appreciate it.
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Well when I equipped my helm from Crota's End the Speaker didn't have any radiant materials for exchange.
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It doesn't matter if you have the gear on, you have to have the materials you wish to exchange in your inventory when you go to him. If you go to the speaker with no radiant materials on you then they wont show up on his list of items to exchange.