The radical 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

$$\ell_f(N)=\operatorname{rad}\operatorname{lcm}(f(1),\ldots,f(N)),$$

where $\operatorname{rad}(n)$ is the product of the distinct primes dividing $n$. Radical LCM conjecture. As $N\to\infty$,

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

The paper proves the lower bound $\log\ell_f(N)\gtrsim\frac{2}{d}N\log N$. The conjecture asserts that the radical of the least common multiple has the same leading asymptotic as the least common multiple itself, but this remains open.