Left-definite domain equality conjecture for Sturm–Liouville operators

Let ${\bf L}^n$ be a self-adjoint operator defined by left-definite theory on $L^2[(a,b),w]$ with domain $\mathcal{D}_{\bf L}^n$ that includes a complete system of orthogonal polynomial eigenfunctions. Let $\mathcal{A}_n$, $\mathcal{B}_n$, $\mathcal{C}_n$, and $\mathcal{F}_n$ be the domains defined by the maximal-domain, boundary-form, and Lagrangian boundary conditions described above, and suppose that $\mathcal{D}_{\bf L}^n=\mathcal{A}_n$. The operator acts through the order-$2n$ differential expression $\ell^n[\,\cdot\,]$, generated by composing a Sturm–Liouville differential operator with itself $n$ times, and extends the minimal operator $L^n_{\scriptstyle \text{min}}$ with deficiency indices $(n,n)$. Left-definite domain equality conjecture. For every $n\in\mathbb{N}$,

$$\mathcal{A}_n=\mathcal{B}_n=\mathcal{C}_n=\mathcal{F}_n=\mathcal{D}_{\bf L}^n.$$

The conjecture was answered affirmatively for the Jacobi differential operator, while its validity for other operators remains open, particularly when there is no complete system of orthogonal polynomial eigenfunctions.