The straightening theorem for irreducible continuous lattices with minimal elements

Let $L$ be an irreducible and continuous lattice with minimal elements, and let $n$ be the cardinality of one of its affine references. An involutory anti-automorphism is a map $\perp:L\rightarrow L$ that is order-reversing and satisfies $\perp^2=\operatorname{id}_L$. Straightening theorem. There exists an involutory anti-automorphism $\perp:L\rightarrow L$ such that $(L,\leq,\perp)$ is a factorial lattice of type $I_n$. The conjecture proposes that these lattice-theoretic axioms suffice to recover the orthocomplementation and factorial $W^*$-lattice structure associated with a finite-dimensional Hilbert lattice; the source presents it as the strongest result sought and does not report a proof.