I don't have a full answer. I hope that my observation provides some insight and a starting point. Let all vertices have weight $\ 1.\ $ Consider an arbitrary plane $\ W\ $ which contains face $\ F\ $ of $\ P_n\ $ (in particular of $\ P_1).\ $ Then plane $\ W\ $ contains the respective faces of all next generation polyhedrons, and the center of vertices of $\ F,\ $ and of all next generation faces contained in $\ W\ $ is the same point belonging to the respective polyhedrons $\ P_{n+d}\ (d=0\ 1\ \ldots)$. Thus all the mentioned centers belong to the topological boundary of intersection $\ \bigcap_{n=1\ 2\ \ldots} P_n$.

I don't have a full answer. I hope that my observation provides some insight and a starting point. Let all vertices have weight $\ 1.\ $ Consider an arbitrary plane $\ W\ $ which contains face $\ F\ $ of $\ P_n\ $ (in particular of $\ P_1).\ $ Then plane $\ W\ $ contains the respective faces of all next generation polyhedrons, and the center of vertices of $\ F,\ $ and of all next generation faces contained in $\ W\ $ is the same point belonging to the respective polyhedrons $\ P_{n+d}\ (d=0\ 1\ \ldots)$. Thus all the mentioned centers belong to the topological boundary of intersection $\ \bigcap_{n=1\ 2\ \ldots} P_n$.

Furthermore (an EDIT session), each such center is the only point of the respective plane $\ W.\ $ This is a very strong indication that the intersection $\ \bigcap_{n=1\ 2\ \ldots} P_n\ $ is strictly convex.

I don't have a full answer. I hope that my observation provides some insight and a starting point. Let all vertices have weight $\ 1.\ $ Consider an arbitrary plane $\ W\ $ which contains face $\ F\ $ of $\ P_n\ $ (in particular of $\ P_1).\ $ Then plane $\ W\ $ contains the respective face of all next generation polyhedrons, and the center of vertices of $\ F,\ $ and of all next generation faces contained in $\ W\ $ is the same point belonging to the respective polyhedrons $\ P_{n+d}\ (d=0\ 1\ \ldots)$. Thus all the mentioned centers belong to the topological boundary of intersection $\ \bigcap P_{n=1\ 2\ \ldots}$.

I don't have a full answer. I hope that my observation provides some insight and a starting point. Let all vertices have weight $\ 1.\ $ Consider an arbitrary plane $\ W\ $ which contains face $\ F\ $ of $\ P_n\ $ (in particular of $\ P_1).\ $ Then plane $\ W\ $ contains the respective faces of all next generation polyhedrons, and the center of vertices of $\ F,\ $ and of all next generation faces contained in $\ W\ $ is the same point belonging to the respective polyhedrons $\ P_{n+d}\ (d=0\ 1\ \ldots)$. Thus all the mentioned centers belong to the topological boundary of intersection $\ \bigcap_{n=1\ 2\ \ldots} P_n$.