A family of Lempert domains

In \cite{G-Z} G.~Ghosh and W. Zwonek introduced a new class of domains $\bL_n$, $n\ge1$, 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 $\bL_n$ are Lempert domains.