A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Let $G$ be a locally compact group, and let $\mathcal{WAP}(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[\operatorname{SIN}]$-group, but not compact, then the dual Banach algebra $\mathcal{WAP}(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[\operatorname{SIN}]$-group, $\mathcal{WAP}(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups $G$ such that $\mathcal{WAP}(G)^\ast$ does have a normal, virtual diagonal.