  A conjecture due to Cilleruelo states that for an irreducible polynomial $f$
with integer coefficients of degree $d\geq 2$, the least common multiple
$L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log
L_f(N)\sim (d-1)N\log N$ as $N\to \infty$. We establish a version of this
conjecture for almost all shifts of a fixed polynomial, the range of $N$
depending on the range of shifts.
