Buzzard–Kilford boundary eigencurve conjecture for ramified nebentypus slopes

Let $p$ be an odd prime, let $N$ be coprime to $p$, and let $\psi_0$ be a character of $(\mathbb{Z}/p\mathbb{Z})^\times$ with $\psi_0(-1)=-1$. For $m\geq 2$, let $\psi$ be a character of $(\mathbb{Z}/p^m\mathbb{Z})^\times$ of exact conductor $p^m$, and let $k+1\geq 2$ be an integer. Write $\omega$ for the Teichmüller character. Buzzard–Kilford's conjecture. There exists a non-decreasing sequence of rational numbers $a_1,a_2,\dots$ tending to infinity such that, whenever

$$\psi|_{(\mathbb{Z}/p\mathbb{Z})^\times}\cdot\omega^k=\psi_0,$$

the slopes of $U_p$ on $S_{k+1}(\Gamma_0(p^mN);\psi)$ are the first terms of

$$a_1/p^m,\ a_2/p^m,\ \dots$$

consisting of all terms strictly less than $k$ and some terms equal to $k$. Moreover, the sequence $a_1,a_2,\dots$ is a union of finitely many arithmetic progressions. This is an expectation about the eigencurve near the boundary of weight space; the paper presents its slope formula as positive evidence, while the general assertion remains open.