Sharp inclusion conjecture for triangular ratio and hyperbolic metric balls

Let $G=\mathbb{B}^n$, and let $B_s(x,t)$ and $B_\rho(x,R)$ denote the balls of radii $t$ and $R$ centered at $x$ in the triangular ratio metric and the hyperbolic metric, respectively. For $0<t<1$, define

l(t,|x|)=\max\left\\{(1+t)(1+|x|)(1-3t+|x|(1+t)),\\ (1-t)(1-|x|)(1+3t-|x|(1-t))\right\\}.

Sharp inclusion conjecture. For $x\in G$ and $R>0$, $B_s(x,t_0)\subseteq B_\rho(x,R)\subseteq B_s(x,t_1)$ if and only if

$$0<t_0\leq\frac{\operatorname{th}(R/2)(1-|x|^2)}{2(1-|x|\operatorname{th}(R/2))-|2|x|-\operatorname{th}(R/2)(1+|x|^2)|}$$

and

$$t_1\geq\frac{(1+|x|)(e^R-1)}{3+e^R+|x|(e^R-1)}.$$

Moreover, for $x\in G$ and $0<t<1$, $B_\rho(x,R_0)\subseteq B_s(x,t)\subseteq B_\rho(x,R_1)$ if and only if

$$0<R_0\leq\log\left(1+\frac{4t}{(1-t)(1+|x|)}\right)$$

and

$$R_1\geq 2\operatorname{arsh}\left(\frac{2t}{\sqrt{l(t,|x|)}}\right).$$

The conjecture concerns sharp radii for mutual inclusions between triangular ratio metric balls and hyperbolic metric balls in the unit ball. The preceding results establish related one-sided bounds, and computer tests suggest that these exact bounds hold; proving the remaining inclusions would settle the conjecture.