Optimal $delta$ for Gromov’s $delta$-hyperbolicity of the hyperbolic plane

What is the minimal $delta$ such that the hyperbolic plane is $delta$-hyperbolic, in the sense of the four point definition of Gromov?

Four point definition of Gromov: A metric space $(X, d)$ is $delta$-hyperbolic if, for all $w, x, y, z in X$,
$$ d(w, x) + d(y, z) leq text{max}{d(x, y) + d(w, z), d(x, z) + d(w, y) } +2delta. $$

Empirically, the minimal value seems to be approximately $0.693$.

There is a related question, but this concerns the optimal $delta$ in the $delta$-slim definition. While this implies a bound on the $delta$ of the four point definition, it hasn’t yet helped me to derive the minimal value.

Any help (or a reference) would be greatly appreciated!