Further remarks on the higher dimensional Suita conjecture

For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)\lambda (I^k_D(z))$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda (I^k_D(z))$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. This biholomorphic invariant was introduced by Błocki. We study its limiting boundary behaviour on two classes of domains: $h$-extendible and strongly pseudoconvex polyhedral domains.