I appreciate you chiming in Dave, as I know you have some experience with these in general. It did, indeed seem a sensible idea for durability.
I cut the cork as I would ordinarily, and then I measured the "box" (10mm in one direction and 11 mm in the other), took a mechanical pencil to lightly scribe my rudimentary cuts, and then I used a small Xacto blade to open up the box. I test fit, and made subtle modifications until it straddled the box well enough to hold seal. It took me three tries, (and the photo above is of attempt #2, as the cork was a hair too loose), and the top portion of the cork above the box is only 2mm deep (wide?) so I anticipate having to revisit this with greater frequency than a typical cork.