The separable straightening theorem for irreducible continuous lattices

Let $L$ be an irreducible, complete and continuous lattice such that every element is the supremum of a family of minimal elements and $1$ is the supremum of a countable family of minimal elements. An involutory anti-automorphism is a map $\perp:L\rightarrow L$ that is order-reversing and satisfies $\perp^2=\operatorname{id}_L$. Separable straightening theorem. There exists an involutory anti-automorphism $\perp:L\rightarrow L$ such that $(L,\leq,\perp)$ is a factorial lattice of type $I_{\omega}$. This is the separable analogue of the finite-type straightening claim; the source states that it can be conjectured after giving the uniqueness of the factorial $W^*$-lattice of type $I_{\omega}$, but gives no proof or resolution.