Sharp Cassinian-to-$j$-metric ball inclusion conjecture

Let $D\subsetneq\mathbb{R}^n$ be a proper subdomain, let $x\in D$, and let $\delta_D(x)$ be the Euclidean distance from $x$ to $\partial D$. For a metric $m$, write $B_m(x,s)=\{y\in D:m_D(x,y)<s\}$. Sharp Cassinian-to-$j$-metric inclusion conjecture. For $t>0$, one has

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

where

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

Moreover, the radii $r$ and $R$ are best possible and $R/r\to1$ as $t\to0$. This generalizes the proved punctured-domain inclusion, but remains conjectural for arbitrary proper subdomains.