Reply by Sark

Plywood can certainly be routered. Done all the time. But quality bits make a big difference. For your project, a downcut bit is what you need. You can lose some control of the router when you encounter an air pocket or void, and that can totally muck up the final product. And its possible that the void is quite visible when you’re finished, and the voids would need to be filled to look right. So buy a piece of quality plywood that is doesn’t have voids if you can.