Gouvêa's limiting-measure conjecture for slopes of p-oldforms

Fix a prime $p$ and an integer $N\geq 1$ coprime to $p$. For each weight $k\geq 2$, consider the multiset of slopes of the $p$-oldforms in $S_k(\Gamma_0(Np))$, normalized by dividing each slope by $k-1$, and form the associated probability measure on $[0,1]$.

Gouvêa's limiting-measure conjecture. As $k$ tends to infinity, these slopes converge to the measure that is uniform on

$$[0,\tfrac{1}{p+1}]\cup[\tfrac{p}{p+1},1]$$

and is zero elsewhere.

The conjecture concerns the asymptotic distribution of normalized slopes as the weight grows; the source poses the existence and form of the limiting measure as an open question.