A family of Lempert domains

In \cite{G-Z} G.~Ghosh and W. Zwonek introduced a new class of domains ${\text{bL}}_n$, $n\ge 1$, which are 2-proper holomorphic images of the Cartan domains of type four. This family contains biholomorphic images of the symmetrized bidisc and the tetrablock. It is well-known, that symmetrized bidisc and tetrablock are Lempert type domains. In our paper we show that the whole family of domains ${\text{bL}}_n$ are Lempert domains.