Friedrichs extension and powers conjecture

Let $A$ be a closed semi-bounded symmetric operator, let ${\bf A}$ denote the associated operator, and let ${\bf A}_F$ be its Friedrichs self-adjoint extension. For $r\in\mathbb{N}$, compare the $2r$-th left-definite space of ${\bf A}_F$ with the domain of the Friedrichs extension of the $r$-th power of ${\bf A}$. Friedrichs powers conjecture.

$$\operatorname{dom}(({\bf A}_F)^r)=\operatorname{dom}(({\bf A}^r)_F).$$

The equality holds in the Jacobi differential-operator case, and every computed case mentioned in the source supports it, but its status in the stated generality is unclear.