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I just realized that my earlier piece count estimate was based on the sets that were visible in the seller's pictures, but now that I've spent six hours or so disassembling sets, I'm realizing that it's a bit more than that. Adding up all the part counts of the instruction books (which all seem to have complete sets) gives a total of 6,727.
For perspective, this is what I've got disassembled so far, and it's about 8 inches deep.