Cilleruelo's least-common-multiple conjecture for polynomial sequences

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define

$$L_f(N)=\operatorname{lcm}\{f(n):1\le n\le N\},$$

ignoring zero values and taking the least common multiple of an empty set to be $1$. Cilleruelo's conjecture. As $N\to\infty$,

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

Cilleruelo proved the conjecture when $d=2$. The paper gives the lower bound $\log L_f(N)\gtrsim N\log N$, while the conjectured asymptotic remains open for general degree $d\ge 2$.