Larry,
Thank you for this provocative thread with useful information for the serious collector. I delayed posting this response until all my boxes were inverted.

I can happily report that the inverted box has the same volume as its pre-inverted cousin.
I would add that with the BCW box shown in your example care must used to remove the end flap from the slot. If it is bent in removing it, getting it to stay in place in the inverted configuration may be difficult.
Upward (toward the box top) pressure on the inside of the end flap is useful in sliding this flap out of its slot without damaging it.
I can also report that the technique works with the "Brockelman" box, which has a different shape from the BCW box when deconstructed.
Of course this inversion process takes time and time is money, so selling on BST and shipping in inverted boxes will necessarily involve a "handling" premium. I am sure that buyers of inverted boxes will understand the necessity for this.
I know that neither you or I will be in Atlantic City this year, but we have a full year to arrange for a competition in Chicago to crown the champion "box inverter". I suspect that Guinness would be happy to attend to certify the champion's time and include it in the next issue of their World Records.
The international appeal of this technique would also open the door for future Olympic consideration.
I would encourage readers of this thread to scour the BST listings and make a purchase from me to become my first inverted box customer.
In fact as a special offer to celebrate this thread, I will send any board member an empty inverted box for the shipping and handling fee only.