Conjecture on the distribution of slopes of classical p-oldforms

Let $p$ be a prime, $N$ a positive integer relatively prime to $p$, and $k$ an integer. Let ${\mathcal M}_k(N,{\bf Z}_p)$ denote the relevant space of classical modular forms, and let $R_k$ be the multiset, with multiplicity, of slopes of classical $p$-oldforms in this space.

Slope-distribution conjecture. The probability that an element of $R_k$ chosen with uniform distribution lies in the interval

$$\left(\frac{k-1}{p+1},\frac{p(k-1)}{p+1}\right)$$

\ndiminishes to zero as $k$ increases without bound.

This conjecture concerns the asymptotic distribution of slopes of classical modular forms. The source presents it as a conjecture from an external reference and reports evidence for it, but gives no resolution.