Problem with kitchen casing/ventilation

Hello forum, I removed the boxing on the top side of the kitchen up to the ceiling today. The boxed-in space turns out to be a great resonance chamber for the supply air ventilation. The ventilation grille is close to the kitchen and the pipes run flush under the plaster straight "through" the boxing into the technical room located behind the kitchen. Attached is a photo of the boxing and the ventilation grille. The boxing consists of MDF boards attached to wooden blocks, which in turn are fixed to the ceiling and the top of the cabinets. Actually pretty normal, I would say. What can I do to dampen the increased ventilation noise? Stuff or line the boxing space? (Sorry for the poor lighting. I’m only concerned about the location of the ventilation).

Insulate (with as dense/heavy as possible) stone wool and structurally decouple.

Thin rubber pads under all fastening points of the case on the substructure and at the fastening points of the exhaust. If that is not enough: coat the case from the inside with bitumen mats from the automotive sector.