There is a persistent 'myth' about counter balanced looms. Well, two, actually.
One is that counter balanced looms cannot weave 'unbalanced' weaves. The other is that they can only be four shafts.
I have tried to explain on chat groups and such that this information is not correct. These comments are more correct for roller type counter balanced looms, but completely incorrect for looms with pulley and lever systems.
This morning I pulled Jane Evans' book A Joy Forever off the shelf to look for some information. When I opened it, the paperback book spine finally broke completely. Since the pages are 8.5 x 11", I grabbed my box of plastic pocket pages and a binder and started carefully pulling the pages apart and inserting them into the pocket pages.
As I was doing this, the above diagram caught my eye. I have been using Laila Lundell's Big Book of Weaving diagram showing how counter balanced looms with horses can be 6, 8, even 10 shafts. This one shows a loom with pulleys with 16 shafts (on the left) and 10 (on the right).
Ten shaft looms were routinely used (and still are in Scandinavia - and I assume Latvia and other eastern European countries) for weaving cloth of two satin blocks. A 10 shaft loom can do two 5 end satin blocks. By their very nature, satin weaves are 'unbalanced'.
So yes, counter balanced looms can be more than four shafts. And yes, counter balanced looms can weave unbalanced weave structures.
The constraints in the 'myth' are for roller type looms, not looms with pulleys and/or 'horses' (levers).
For anyone wanting to look this up - A Joy Forever by Jane Evans, pages 17 and 18.