Let $k$ be a number field, and let $G$ be a split simply connected algebraic group over $k$. Let $\Omega_k$ denote the set of places of $k$. Let $T$ be a maximal torus of $G$ (defined over $k$). Consider the second Tate-Shafarevich group $$ Ш^2(k,T)=\ker\left[H^2(k,T)\to\prod_{v\in\Omega_k} H^2(k_v,T)\right].$$

Question.Is it true that $Ш^2(k,T)=1$ for any maximal torus $T$ of any split simply connected $k$-group $G$?

Note that the answer is YES for $G=SL_n$. Indeed, then $T=\ker[N: S\to {\mathbb G}_m ]$, where $S$ is a maximal torus in $GL_n$. We have a cohomology exact sequence $$ 1\to H^2(k,T)\to H^2(k,S)\to H^2(k,{\mathbb G}_m)$$ and similar exact sequences for any place $v$ of $k$. We see that $Ш^2(k,T)$ embeds into $Ш^2(k,S)$. Since $S$ is a product of tori of the form $R_{L_i/k}\, {\mathbb G}_{m,L_i}$ where each $L_i$ is a finite extension of $k$, we have $$ Ш^2(k,S)=\prod_i Ш^2(k,R_{L_i/k}\,{\mathbb G}_{m,L_i})=\prod_i Ш^2(L_i,{\mathbb G}_m)=1, $$ hence $Ш^2(k,T)=1$.

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