Generically transitive actions on multiple flag varieties

Let $G$ be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type $A$, and $P\subseteq G$ be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples $(G,P,n)$ such that (a) there exists an open $G$-orbit on the multiple flag variety $G\mathrm{/}P\times G\mathrm{/}P\times \dots \times G\mathrm{/}P$ ($n$ factors), (b) the number of $G$-orbits on the multiple flag variety is finite.