Metric comparison conjecture for $k$-metrics on $\ell^s$- and $\ell^t$-type domains

For $0<s<t$, let $G_s$ and $G_t$ be the planar domains

$$G_s=\{(x,y): |x|^s+|y|^s\leq 1\},\qquad G_t=\{(x,y): |x|^t+|y|^t\leq 1\}.$$

For points $z_1,z_2\in G_s$, write $k_{G_s}(z_1,z_2)$ and $k_{G_t}(z_1,z_2)$ for their quasihyperbolic distances in the respective domains. The metric comparison conjecture. For all $z_1,z_2\in G_s$,

$$k_{G_s}(z_1,z_2)\geq 2^{\frac{1}{s}-\frac{1}{t}}k_{G_t}(z_1,z_2).$$

This conjecture generalizes the preceding comparison results for the cases $t=1$ and related power-type domains. Its status is not resolved in the supplied source context.