Unramified cohomology of classifying varieties for exceptional simply connected groups

Let $BG$ be a classifying variety for an exceptional simple simply connected algebraic group $G$. We compute the degree 3 unramified Galois cohomology of $BG$ with values in $(\mathbb{Q} /\mathbb{Z} )'(2)$ over an arbitrary field $F$. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for $G$ semisimple simply connected over all fields.

These computations provide another family of examples of simple simply connected groups $G$ such that $BG$ is not stably rational.