  Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and
$\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let
$K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class
field. For a class group character $\chi$ over $K$, let $\mathbb{Q}(\chi)$ be
the field generated by the image of $\chi$ and $\mathfrak{p}_{\chi}$ the prime
of $\mathbb{Q}(\chi)$ above $p$ determined via $\iota_p$. Under mild
hypotheses, we show that the number of class group characters $\chi$ such that
the $\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with
trivial $\mathfrak{p}_{\chi}$-part increases with the absolute value of the
discriminant of $K$.
