Proportion conjecture for chi-regular primes

Let $\chi$ be a Dirichlet character, and let $\operatorname{ord}(\chi)$ denote its order. A prime $p$ is $\chi$-regular when the relevant even twists $\chi\omega^i$ have vanishing Iwasawa lambda-invariant, with the corrected invariant used when $\chi\omega^i(p)=1$. Proportion conjecture for chi-regular primes. For a fixed character $\chi$, the proportion of $\chi$-regular primes is

$$1+\frac{e^{-1/2}-1}{\varphi(\operatorname{ord}(\chi))},$$

while the proportion of $\chi$-irregular primes is

$$\frac{1-e^{-1/2}}{\varphi(\operatorname{ord}(\chi))}.$$

For the trivial character this gives the familiar predicted proportion $e^{-1/2}$ of regular primes; the prediction is derived from the preceding random-matrix heuristic and is supported by numerical data.