In Cassels's 1962 ICM paper (available here), he says the following: "Indeed, Tate and Šafarevič have, I believe, independently conjectured(5) that Ш itself is always finite", and the (5) is a footnote stating: "In his lecture, Tate denied paternity but adopted the conjecture. In conversation during the Congress Šafarevič expressed strong doubts."

So, maybe no one knows!

In Cassels's 1962 ICM paper (available here), he says the following: "Indeed, Tate and Šafarevič have, I believe, independently conjectured(5) that Ш itself is always finite", and the (5) is a footnote stating: "In his lecture, Tate denied paternity but adopted the conjecture. In conversation during the Congress Šafarevič expressed strong doubts."

So, maybe no one knows!

EDIT: As an added bonus, I found the following quote in Cassels's review of Silverman's book: "Without doubt the reviewer's most lasting contribution to the theory is the introduction of the Cyrillic letter Ш ("sha") to denote this group, a usage which has become universal."