The endomorphism conjecture for F-roots of the full twist

Let $B$ be the braid group, let $B^+$ be its positive braid monoid, let $F$ act on $B$ by the Frobenius diagram automorphism, and let ${\boldsymbol\pi}$ be the positive generator of the center of the pure braid group. Let ${\mathcal B}$ and ${\mathcal D}$ be the categories defined by $F$-conjugation of braids, so that $\operatorname{End}_{\mathcal B}({\bf b})=C_B({\bf b}F)$ and $\operatorname{End}_{\mathcal D}({\bf b})$ is obtained by inverting the morphisms in ${\mathcal D}^+$. An element ${\bf b}\in B^+$ is an $F$-root of ${\boldsymbol\pi}$ if $({\bf b}F)^d={\boldsymbol\pi}F^d$ for some positive integer $d$. Endomorphism conjecture. When ${\bf b}$ is an $F$-root of ${\boldsymbol\pi}$,

$$\operatorname{End}_{\mathcal D}({\bf b})=\operatorname{End}_{\mathcal B}({\bf b})=C_B({\bf b}F).$$

The paper proves this in several cases, including ${\bf b}={\boldsymbol\pi}$, ${\bf w}_0$, Coxeter elements, roots in types $A$ and $B$, and fourth roots in type $D_4$, but does not establish it in general.