Cilleruelo's LCM growth conjecture for irreducible polynomial sequences

Let $f\in\mathbb Z[X]$ be an irreducible polynomial with degree $\deg f\geq 2$, and define

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

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

$$\log L_f(N)\sim (\deg f-1)N\log N.$$

This conjecture predicts that the least common multiple of the first $N$ values of an irreducible polynomial of degree at least two grows faster than linearly. It is known for degree $2$, while the general case remains open; the paper proves the lower bound $\log L_f(N)\gg N\log N$ for all degrees $\deg f\geq 2$.