An improved bound on the least common multiple of polynomial sequences

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ of degree $d\ge 2$ is irreducible over the rationals, then $\log\operatorname{lcm}(f(1),\ldots,f(N))\sim(d-1)N\log N$ as $N\to\infty$. He proved it for the case $d = 2$. Very recently, Maynard and Rudnick proved there exists $c_d > 0$ with $\log\operatorname{lcm}(f(1),\ldots,f(N))\gtrsim c_d N\log N$, and showed one can take $c_d = \frac{d-1}{d^2}$. We give an alternative proof of this result with the improved constant $c_d = 1$. We additionally prove the bound $\log\operatorname{rad}\operatorname{lcm}(f(1),\ldots,f(N))\gtrsim\frac{2}{d}N\log N$ and make the stronger conjecture that $\log\operatorname{rad}\operatorname{lcm}(f(1),\ldots,f(N))\sim (d-1)N\log N$ as $N\to\infty$.