Sharp Cassinian-to-quasihyperbolic ball inclusion conjecture in a punctured space

Let $a\in\mathbb{R}^n$, let $D=\mathbb{R}^n\setminus\{a\}$, let $x\in D$, and let $B_k(x,s)$ and $B_c(x,s)$ denote quasihyperbolic and Cassinian metric balls, respectively. Sharp Cassinian-to-quasihyperbolic inclusion conjecture. For $t>0$,

$$B_k(x,r)\subset B_c(x,t)\subset B_k(x,R),$$

where

$$r=\log(1+t|x-a|),\qquad R=\log\left(\frac{1}{1-t|x-a|}\right).$$

The radii $r$ and $R$ should be sharp, with $R/r\to1$ as $t\to0$. The inner inclusion is known in the paper, while sharpness and the outer inclusion are posed as a conjecture.