Carve the parts.
You need one of part 1 – the rectangular piece,
and two of part 2 – the lozenge shape.
You’ll need to adjust the cutting depth to match your material of course.
I used 6mm ply, but it actually measured 5.7mm thick, and the Easel file worked for that.
What you need is for the grooves to be deep enough to make part 1 flexible enough, when wet, to wrap right around, but not so deep that it falls apart!
The outline cut depth I used left a little material to prevent the part flying out of the machine. This is just cut with a craft knife afterwards to remove the parts.
To keep the top and bottom together when the box is closed, I’ve glued in a hardwood lip.
Soaking the thin hardwood strips in boiling water allows them to be formed around a slightly tighter radius than the box requires. Tape them in place and allow to cool.
Once released, the bent strips will open a little, but they should easily be pliable enough to glue into place in the top of the box’s bottom half. Leave about ¼" (6mm) extending out, for the box lid to fit onto.
Clamp and leave to cure.