Infinitude conjecture for nontrivial relative torsion groups in cyclotomic fields

Let $N\geq 2$ and let $p$ be a prime with $p\nmid N$. Let $\psi_N$ be a character of $\mathbb{Q}(\mu_N)$ of order $N$, and let ${\mathfrak p}_N$ be a prime above $p$ in $\mathbb{Q}(\mu_N)$. Infinitude conjecture for relative torsion groups. There exist infinitely many pairs $(N,p)$ such that

$$\frac{1}{2}L_p(1,\psi_N)\equiv 0\pmod{{\mathfrak p}_N},$$

so that ${\mathcal T}_{\mathbb{Q}(N)}^*\ne 1$. This predicts infinitely many nontrivial relative torsion groups when the parameters $N$ and $p$ vary; for fixed $K\subset\widehat{\mathbb{Q}}$, the source separately mentions a conjectured finiteness statement.