Can you prove or disprove the following claim?

Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the inscribed circle and polygon sides such that $B_1$ lies on $A_1A_2$, $B_2$ lies on $A_2A_3$ ,etc. Denote by $H_1,H_2, \ldots,H_n$ the orthocenters of the triangles $\triangle A_1B_1B_n$, $\triangle A_2B_2B_1$,....,$\triangle A_nB_{n}B_{n-1}$ . Then the polygon is cyclic if and only if $H_1,H_2;\ldots ,H_n$ are concyclic.

Picture for the case $n=6$:

GeoGebra applets that demonstrate this claim can be found here , here and here.