Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be

$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \not \in S}H^1(k_v,E_v)\right).$$

Note that the normal Tate-Shafarevich group is $Ш(E) = Ш(E,\emptyset)$. Recall that the Tate-Shafarevich conjecture states that $Ш(E)$ is finite. My question concerns the corresponding problem for $Ш(E,S)$.

Is $Ш(E,S)$ conjecturally finite?

If the answer to this is yes, then an obvious next question is whether this is equivalent to finiteness of $Ш(E)$, i.e.

Is it known that $Ш(E) \subset Ш(E,S)$ has finite index?

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