I wouldn't have thought they'd fit but I tried it and it works, barely, with a 16" square. 12s would be too small.
You are right, it barely works. It looks a bit stretched. I think the classic cotton bandana size is ideal or a little larger, say 22"-26". Perhaps that is an opportunity for scarf and pocket square suppliers to fill? I know I would like to find a few in cashmere/silk or silk/linen blends in fairly simple colors and patterns.