Gouvêa–Mazur conjecture on local constancy of slopes

Let $p$ be a prime number, let $N$ be a positive integer coprime to $p$, and normalize the $p$-adic valuation on $\overline{\mathbb{Q}}_p$ by $v_p(p)=1$. For $r\in\mathbb{Q}$ and a space $S$ of modular forms with $U_p$-action, let $d(S,r)$ be the number of generalized eigenvalues of $U_p$ with $p$-adic valuation $r$. Let $S_k(\Gamma_0(Np))$ be the space of cusp forms of weight $k$ and level $Np$. Gouvêa–Mazur conjecture. The polynomial $M(x)$ in the preceding local-constancy statement can be chosen as $M(x)=x$; equivalently, for every $r\in\mathbb{Q}_{\geq 0}$, if $k_1,k_2$ are integers satisfying

$$k_1\geq 2r+2,\qquad k_2\geq 2r+2,$$

and

$$k_1\equiv k_2\pmod{(p-1)p^m}$$

for an integer $m\geq r$, then

$$d\left(S_{k_1}(\Gamma_0(Np)),r\right)=d\left(S_{k_2}(\Gamma_0(Np)),r\right).$$

The conjecture concerns the local constancy of slopes of modular forms as the weight varies $p$-adically. Buzzard and Calegari disproved it by giving an explicit counterexample, so the claim is refuted.