Conjecture on equality of the natural domains for left-definite powers

Let ${\bf L}^n$ be a self-adjoint operator defined by left-definite theory on $L^2[(a,b),w]$, with domain $\mathcal{D}_{\bf L}^n$ containing a complete system of orthogonal polynomial eigenfunctions. Suppose ${\bf L}^n$ acts via $\ell^n[\,\cdot\,]$, a differential operator of order $2n$ generated by composing a Sturm--Liouville differential operator with itself $n$ times, where $n\in\mathbb{N}$. Suppose further that ${\bf L}^n$ extends the minimal operator ${\bf L}^n_{\operatorname{min}}$, whose deficiency indices are $(n,n)$.

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 domains are distinct formulations of boundary and regularity conditions for powers of Sturm--Liouville operators. The source notes that the equality is known in some special cases, including the Legendre case for $\mathcal{D}_{\bf L}^n=\mathcal{A}_n$ and the case $n=2$ for all four domains, while the general equality remains elusive.