Horizontal variation of Tate--Shafarevich groups

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota :\bar{\mathbb{Q}}\hookrightarrow {\mathbb{C}}_p$ 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 $K$.