We consider a new class of potentially exotic group C*-algebras $C^*(\text{PF}_{p}^*(G))$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner \cite{BG}. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*(\text{PF}_{p}^*(G))$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*(\text{PF}_{p}^*(G))$. This greatly generalizes earlier results of Okayasu \cite{Okay} and Wiersma on the pairwise distinctness of $C^*_{L^p}(G)$ for $20$ (recall $A_\pi\subseteq B_\pi$). Here $A_\pi$ and $B_\pi$ are the closed span of coefficients of $\pi$ in the norm and $w^*$-topology of the Fourier-Stieljes algebra $B(G)=C^*(G)^*$, respectively. In particular, we have $A_\pi\subseteq B_\pi$.\\ This is based on a joint work with M. Wiersma.