Here's my case size requirement analysis, with the callout that it would be super-helpful to have someone who already has APX to take some measurements to confirm my assumptions:
By scaling and overlaying the silhouette of the APX in the case on Apple's tech spec page with the measurement image, I get an approximate width of around 8.5", and assuming that folding the earcups flat increases the height a little bit, let's say the vertical dimension is 7.65" or so.
This means that the smart case actually makes the packed APX wider than they are tall, though I'm not sure how flexible overhang on the horizontal edges of the smart case is–there may be potential here to do some folding/squeeing and reduce the width down from 8.5".
The remaining question is how thick the APX are when in their case. Assuming the headband is the thickest part in this configuration, and going by the proportions on Apple's tech spec silhouette again, they should be slightly under 2".
The net of all this is a target case size of about 8.6" x 7.75" x 2" (I'm assuming we need a little wiggle room beyond the assumed dimensions). Unfortunately there doesn't seem to be a lot out there that's this tall/wide but also relatively thin.
