On the Hadwiger numbers of starlike disks

The Hadwiger number $H(J)$ of a topological disk $J$ in ${\mathrm{\Re }}^2$ is the maximal number of pairwise nonoverlapping translates of $J$ that touch $J$. It is well known that for a convex disk, this number is six or eight. A conjecture of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike disk is at most eight. A. Bezdek proved that this number is at most seventy five for any starlike disk. In this note, we prove that the Hadwiger number of a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger number of a topological disk $J$ such that $(\text{conv}J)\setminus J$ is connected, is six or eight.