Finite-rank perturbation parametrization for powers of Sturm–Liouville operators

Let ${\bf L}^n_0$, with $n\in\mathbb{N}$, be a self-adjoint extension of the minimal operator $L^n_{\scriptstyle \text{min}}$ on $L^2[(a,b),w]$, with deficiency indices $(m,m)$ where $m\in\{n,2n\}$. Let ${\bf L}^n$ act through the semi-bounded order-$2n$ Sturm–Liouville differential expression $\ell^n[\,\cdot\,]$, obtained by composing the Sturm–Liouville operator with itself $n$ times, and let $\Theta$ be a self-adjoint relation in $\mathbb{C}^m$. Finite-rank perturbation conjecture. There exist choices of ${\bf L}^n_0$, ${\bf B}$, and ${\bf B}^*$ such that

$${\bf L}_{\Theta}:={\bf L}^n_0+{\bf B}\Theta{\bf B}^*$$

is well-defined, has rank-$m$ singular perturbation form, and every self-adjoint extension of the minimal operator $L_{\scriptstyle \text{min}}$ can be written as ${\bf L}_{\Theta}$ for some $\Theta$. This would extend the rank-two perturbation parametrization to powers of Sturm–Liouville operators; no proof or resolution is supplied.