Optimal bounds for the Kobayashi distance near ${\mathcal{C}}^2$-smooth boundary points

It is shown that the optimal upper and lower bounds for the Kobayashi distance near ${\mathcal{C}}^{2,\alpha }$-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: “Precise estimates of invariant distances on strongly pseudoconvex domains”, Adv. Math. 478 (2025), 110388, remain true in the general ${\mathcal{C}}^2$ strongly pseudoconvex setting. In fact, the upper bound is extended to the general ${\mathcal{C}}^{1,1}$-smooth case. We also give upper and lower bounds for the Kobayashi distance near non-semipositive boundary points.