Do gems affect converted moves if the type matches the conversion?

Question

EDIT: I forgot to finish typing it. Dumb me.

For example, since I know Normal Gem doesn't boost moves that are converted via Abilities such as Refrigerate, does an Ice Gem boost a Refrigerated Normal-type attack instead?

Answer 1

I answered a question regarding this same topic, just slightly different.
On this question I state that Normal Gems will not boost Refridgerated/Pixilated/whatevered moves, so this obviously means that the Ice Gem will boost Refridgerated/Pixilated/whatevered moves.

Hope I helped. :)
Sources: Above Link and Logic