Yamazaki's generalized Birch–Tate conjecture for tori

Let $K$ be a totally real number field and let $T$ be a torus over $K$ that is split by a totally real field. Let $X(T)=\operatorname{Hom}(\mathbb{G}_m,T)$ be the cocharacter group, let $L_K(X(T),s)$ be the Artin $L$-function attached to the resulting finite-image representation of the absolute Galois group of $K$, let $W^T(K)=H^0(K,X(T)\otimes\mathbb{Q}/\mathbb{Z}(2))$, and let $K^T(O_K)$ be the subgroup of Somekawa's Milnor $K$-group $K^T(K)=K(K;T,\mathbb{G}_m)$ defined in the source. Generalized Birch–Tate conjecture. The equality

$$|L_K(X(T),-1)|=\frac{|K^T(O_K)|}{|W^T(K)|}$$

should hold. This generalizes the classical Birch–Tate conjecture, since for $T=\mathbb{G}_m$ the three terms specialize to the Dedekind zeta function, $W_2(K)$, and $K_2(O_K)$, respectively. The paper states that it proves the conjecture for a certain class of tori.