Weak smooth-values conjecture for polynomial sequences

For a polynomial $P\in{\mathbb Z}[x]$, let $\Psi_P(x,y)$ count the integers among $\{P(n):1\leq n<x\}$ whose prime factors are all at most $y$. Weak smooth-values conjecture. For every non-constant polynomial $P\in{\mathbb Z}[x]$,

$$\limsup_{x,y\geq2}\frac{\Psi_P(x,y)\log y}{y}=\infty.$$

This weaker conjecture is introduced because the stronger smooth-values conjecture is not sufficient for the paper's application, even under Schinzel's hypothesis $H$. It is known for linear polynomials and, according to the source, in the quadratic case via results of Dartyge–Martin–Tenenbaum.