Linearity of finite simple CKL-algebras

Prove that every finite simple CKL-algebra is linear, thereby extending the classification of simple linear L-algebras to all finite simple CKL-algebras.

Background

The paper introduces the finite CKL-algebras $\mathbf{A}_n$ and proves that every simple linear L-algebra of size $n$ is isomorphic to $\mathbf{A}_n$. It also defines tail$^{+}$ CKL-algebras and establishes that every simple tail$^{+}$ CKL-algebra is linear, hence isomorphic to one of the $\mathbf{A}_n$.

The authors report that computational evidence motivates extending this result beyond the tail$^{+}$ subclass to all finite simple CKL-algebras. The unresolved issue is whether simplicity alone forces the underlying partial order of a finite CKL-algebra to be linear.