Finiteness conjecture for Bloch–Kato Shafarevich–Tate groups

Let $K$ be a number field and let $\text{Ш}^{\operatorname{BK}}(K,\mathcal{M})=\bigoplus_p\text{Ш}_p^{\operatorname{BK}}(K,\mathcal{M})$ be the Bloch–Kato Shafarevich–Tate group of the modular motive. Bloch–Kato Shafarevich–Tate finiteness conjecture. For all number fields $K$, the group $\text{Ш}^{\operatorname{BK}}(K,\mathcal{M})$ is finite. This generalizes the classical finiteness conjecture for Shafarevich–Tate groups of abelian varieties and is open in the stated generality.