I don't know if that's worth it, but it's your house. If there's a bathroom and a closet on either side I wouldn't care too much. Plus just dumping insulation in there isn't going to sound proof much. Especially considering there's probably more shared walls between rooms than that tiny chase.
Also, they make make a fire rated spray foam. It is reddish when sprayed.
That is an odd space, maybe the builders originally intended it as an hvac chase or something. What's under it? You do have a rare opportunity to easily pass things from 1st floor to attic if it's a two story house. Wires, pipe, ducts, etc.
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