Obstructions to a small hyperbolicity in Helly graphs

It is known that for every graph $G$ there exists the smallest Helly graph $\cal H(G)$ into which $G$ isometrically embeds ($\cal H(G)$ is called the injective hull of $G$) such that the hyperbolicity of $\cal H(G)$ is equal to the hyperbolicity of $G$. Motivated by this, we investigate structural properties of Helly graphs that govern their hyperbolicity and identify three isometric subgraphs of the King-grid as structural obstructions to a small hyperbolicity in Helly graphs.