I used a layer of heavy 2mm butyl sound deadening mat to load the panels with mass, with another layer of 40mm convoluted acoustic foam, on all five panels.
I also added foam strips on two facets of every removable panel edge, to form a tight seal and remove any gaps.
After doing this, most of the noise was escaping from the roof-mounted 4x 120mm fan tray, reflecting off the ceiling and reverberating around the room. To mitigate this, I built a quick & dirty plywood baffle/shroud, also lined with butyl mat and foam, that fits over the roof fan grilles. This almost entirely eliminates noise from the fan tray.
The gap between the floor and bottom of the cabinet was sealed with cut-to-size chunks of foam to stop sound escaping from the bottom fan tray, with convoluted foam laid on the floor facing upwards to further dissipate high-frequency reflections against the cabinet bottom.
All of this cost hardly anything, and has made a night-and-day difference to perceived sound levels. Three ear-splittingly loud datacentre switches fitted into this cabinet entirely disappear into the background when the door is closed, with the noise floor from the cabinet being way below the aircon fans.
Crucially, all distressing high-frequency whine is completely gone with the door closed.