Dickman distribution conjecture for large Hecke factors

For primes $p\ne\ell$, let $D_p=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})$, let $D_p^\pm=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})^\pm$, and define

$$Y_\ell(p)=\frac{1}{D_p}\deg_{\mathbb{F}_p}^*\widetilde{\Phi}_{p,\ell}(X),\qquad Y_\ell^\pm(p)=\frac{1}{D_p^\pm}\deg_{\mathbb{F}_p}^*\widetilde{\Phi}_{p,\ell}^\pm(X),$$

where the tilded polynomials are the characteristic polynomials of $T_\ell$ on the indicated spaces. Let $\rho$ be the Dickman function, defined by $\rho(u)=1$ for $u\in(0,1]$ and $\rho(u)=u^{-1}\int_{u-1}^u\rho(t)\,dt$ for $u>1$. Dickman distribution conjecture. For any fixed prime $\ell$,

$$\lim_{X\to\infty}\frac{\#\{p\in[X,2X]:p\text{ prime},\ Y_\ell(p)\leq y\}}{\#\{p\in[X,2X]:p\text{ prime}\}}=\rho\left(\frac{1}{2y}\right)^2.$$

This is the precise distributional prediction derived from the random-permutation model; it describes the asymptotic distribution of the normalized largest irreducible Hecke factor.