The $L^p$-exotic group C*-algebra conjecture

For a locally compact group $G$ and $1\leq p\leq\infty$, let $C^*_{L^p}(G)$ and $C^*_{L^{p+}}(G)$ denote the corresponding exotic group C*-algebras, and let $B_{L^p}(G)$ be the associated coefficient-function space. The $L^p$-exotic group C-algebra conjecture.* One has

$$C^*_{L^p}(G)=C^*_{L^{p+}}(G)$$

for every $1\leq p\leq\infty$. Equivalently,

$$B_{L^p}(G)=\bigcap_{\epsilon>0}B_{L^{p+\epsilon}}(G)$$

for every locally compact group $G$ and $2\leq p\leq\infty$. The conjecture is posed as a natural generalization of the Cowling–Haagerup–Howe theorem and concerns whether the $L^p$ construction changes when one passes to exponents arbitrarily slightly larger than $p$. Its resolution is not given in the supplied text.