The least common multiple of polynomial values over function fields

Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb Z[X]$ of degree $d \geq 2$ one has $$ \log \mathrm{lcm} (f(1),\ldots , f(N))\sim (d-1)N\log N $$ as $N \to \infty $. He proved it in the case $d=2$ but it remains open for every