  The algebra $\mathcal H:= H_{1,\nu}(I_2(2m+1))$ of observables of the
Calogero model based on the root system $I_2(2m+1)$ has an $m$-dimensional
space of traces and an $(m+1)$-dimensional space of supertraces. In the
preceding paper we found all values of the parameter $\nu$ for which either the
space of traces contains a~degenerate nonzero trace $tr_{\nu}$ or the space of
supertraces contains a~degenerate nonzero supertrace $str_{\nu}$ and, as
a~consequence, the algebra $\mathcal H$ has two-sided ideals: one consisting of
all vectors in the kernel of the form $B_{tr_{\nu}}(x,y)=tr_{\nu}(xy)$ or
another consisting of all vectors in the kernel of the form
$B_{str_{\nu}}(x,y)=str_{\nu}(xy)$. We noticed that if $\nu=\frac z {2m+1}$,
where $z\in \mathbb Z \setminus (2m+1) \mathbb Z$, then there exist both a
degenerate trace and a~degenerate supertrace on $\mathcal H$.
  Here we prove that the ideals determined by these degenerate forms coincide.
