Puzzle rack for Lauri puzzles

[williams44]

We have a bunch of Lauri puzzles that I wanted to organize in a puzzle rack. I got one from Lakeshore only to find that the puzzles don't fit because they are too thick. Lauri puzzles are the foam ones that sit in a yellow tray. Here's a link to what I'm talking about, but it's hard to see the yellow tray in the picture.

Anyway, does anyone know if they fit in the Melissa and Doug puzzle racks and if so, which one. Or if there's something else out there that would work.

Thanks.