Non-abelian strong Coates-Sinnott conjecture for higher étale cohomology

Let $K/k$ be a Galois extension of number fields with Galois group $G$, let $p$ be an odd prime, let $n>1$, and let $S$ and $T$ be finite non-empty disjoint sets of places of $k$, with $S$ containing the ramified and infinite places, no $p$-adic place in $T$, and with $C_S^T(\mathbb Z_p(n))$ the complex defined in the paper. Non-abelian strong Coates-Sinnott conjecture. One has

$$\theta_S^T(1-n)\in\operatorname{Fitt}_{\mathbb Z_pG}^{\max}\left(H^2\left(C_S^T(\mathbb Z_p(n))\right)\right).$$

This is the higher analogue of the strong Brumer-Stark property. It is intended to yield a non-abelian analogue of the Coates-Sinnott conjecture; the paper notes that the corresponding strong property does not hold in general in degree zero, whereas the higher analogue is conjectured to behave better.