Asymptotic distribution conjecture for twisted Iwasawa lambda-invariants

Let $\chi$ be a fixed Dirichlet character, set $m=\operatorname{ord}(\chi)$, and let $\theta_{\leqslant X}^{(r)}(\chi)$ count primes $p\leqslant X$ for which the corrected $\chi$-twisted Iwasawa lambda-invariant equals $r$. Asymptotic distribution conjecture. As $X$ tends to infinity,

$$\lim_{X\to\infty}\frac{\theta_{\leqslant X}^{(0)}(\chi)}{X/\log X}=1,$$ $$\lim_{X\to\infty}\frac{\theta_{\leqslant X}^{(1)}(\chi)}{\log\log X}=\frac{1}{\varphi(\operatorname{ord}(\chi))},$$

and $\theta_{\leqslant X}^{(r)}(\chi)$ is bounded above when $r\geqslant 2$. This prediction follows from the random matrix model and the prime number theorem, describing the expected scarcity of primes with positive or large twisted lambda-invariant.