Sah's radical conjecture for polynomial least common multiples

Let $f\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$, and define

$$L_f(N)=\operatorname{lcm}(f(1),f(2),\ldots,f(N)),$$

and

$$\ell_f(N)=\operatorname{rad}(L_f(N))=\prod_{\substack{p\mid f(1)f(2)\cdots f(N)\p\ \mathrm{prime}}}p.$$

Sah's conjecture. As $N\to\infty$,

$$\log\ell_f(N)\sim\log L_f(N)\sim (d-1)N\log N.$$

Sah's conjecture asserts that the radical has the same leading asymptotic as the least common multiple, a strengthening of the available lower bounds for both quantities.