Smooth-values conjecture for irreducible polynomials

Let $P\in{\mathbb Z}[x]$ be an irreducible polynomial of degree $d$, and let $\Psi_P(x,y)$ be the number of integers in the set $\{P(n):1\leq n<x\}$ whose prime factors are all at most $y$. Let $\rho$ denote the Dickman function, defined by $\rho(u)=1$ for $0\leq u\leq1$ and $\rho'(u)=-\rho(u-1)/u$ for $u>1$. Smooth-values conjecture. Then

$$\Psi_P(x,y)\sim\rho(du)x,\qquad (x=y^u).$$

This predicts that irreducible polynomial sequences have the same expected density of smooth values as the integers, after the degree-dependent rescaling. The source says that this is a special case of a more general conjecture, that it was proved under a quantitative version of Schinzel's hypothesis $H$ only in a limited range, and that unconditionally it was known only for linear polynomials; it also reports numerical evidence supporting it.