On the Hadwiger numbers of starlike disks

The Hadwiger number H(J) of a topological disk J in R2 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 6 or 8. A conjecture of A. Bezdek and K. and W. Kuperberg says that the Hadwiger number of a starlike disk is at most 8. Bezdek proved that this number is at most 75 for any starlike disk. In this note, we prove that the Hadwiger number of a starlike disk is at most 35. Furthermore, we show that the Hadwiger number of a topological disk J such that (convJ)∖J is connected is 6 or 8.