} \right)$ at a given $\pi _{S} $, as $$\label{00)}  \left|Z_{R_{j} }^{*} \left(\pi _{S} \right)\right|=\sum _{i,k}Z_{R_{j} }^{\left(\pi _{S} \right)} \left(\left(R_{i} ,\sigma _{k} \right)\right) =\left|{\rm {\mathcal S}}_{I}^{\left(\pi _{S} \right)} \left(R_{j} \right)\right|.$$ 