Gouvêa's distribution conjecture for slopes of modular forms

Let $p$ be a prime number, let $N$ be an integer relatively prime to $p$, and let $a_p$ range over the $T_p$-eigenvalues in $S_k(\Gamma_0(N))$. Define

$$X_T:=\left\{\frac{p+1}{k}\,v_p(a_p): a_p\text{ is a }T_p\text{-eigenvalue in }S_k(\Gamma_0(N)),\ k\leq T\right\}.$$

Gouvêa's distribution conjecture. The set $X_T$ is equidistributed on $[0,1]$ as $T$ tends to infinity.

The paper presents this as a well-known conjecture concerning $U_p$-slopes of modular forms, primarily in connection with $p$-oldforms; it is used as contextual background rather than resolved here.