I dont think I understand your question properly?
The Little Americas Cup (originally called ICCT) was and is sailed with C-class cats and rigid wings up til present.
The organizers of the ICCT wanted to change the format of the event from C-class cats (due to lack of competitors) to F-18HT cats. I dont know if the F-18HT class rules permit rigid sails, but I suspect they dont.
The C-class does it's own thing these days, but I dont think any of the events use the name "Little Americas Cup" officially anymore?