Here's my 39 x 17in paipo made of strips of Paulownia which I finished building a few months ago.
It's about 2cm thick. It has just a single layer of 6oz glass top and bottom with a second layer over the base plate. No carbon except for a couple of small patches over the inserts that probably don't do much.
It pretty strong and stiff and is holding up fine. I wouldn't say there is zero flex, but there is zero noticeable flex with the above configuration while riding. I don't think it's strong enough for a surf foil, where you generally stand a lot further forward, and want to be able to pump it. But for a kitefoil like the Stringyfoil above, riding strapless, it's plenty stiff and strong enough.
So to answer your question, it does depend, but I'd say about 18mm is probably the minimum. It goes up if you're riding a surf wing, or riding with straps. Or you'd have to make it stronger with more layers of glass and/or carbon reinforcements.
The area that takes the most strain and is most likely to fail is the bottom of the board exactly at the front of the base plate. I put a rubber pad there consisting of an old car tyre material. Before I did that, with just the baseplate attached to the board directly, the fibreglass was stressing and showing signs of starting to crack after a few sessions. Putting a layer of rubber there seems to have resolved this.
It weighs about 2.6kg. This is my first build. I made a lot of mistakes, but it turned out fine in the end, and gets the job done.