Cohomological concentration and discrete-series archimedean component

Let $G$ be a group such that $G(\mathbb{R})$ admits discrete series, let $\pi^S$ be an automorphic representation that is not $G$-Eisenstein, and write an automorphic cuspidal representation as $\pi\cong\pi_\infty\pi_S\pi^S$. Let $S_K(G)$ be the associated locally symmetric space, let $V_\lambda$ be the coefficient system, and let $q(G)$ be the relevant cohomological degree. Cohomological concentration conjecture. The cohomology localized at $\pi^S$ satisfies

$$H^i(S_K(G),V_\lambda)(\pi^S)=0$$

for $i\neq q(G)$, and $\pi_\infty$ necessarily belongs to the discrete series. This claim concerns the expected concentration of cohomology and the archimedean representation contributing to it; the supplied source does not state whether it has been proved or disproved.