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 $w, x, y, z \in X$, $$ d(w, x) + d(y, z) \leq \text{max}\{d(x, y) + d(w, z), d(x, z) + d(w, y) \} +2\delta. $$

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!