Function-field analogue of Cilleruelo's least-common-multiple conjecture

Let $q=p^k$ be a prime power, and let $f\in\mathbb{F}_q[T][X]$ be a fixed irreducible polynomial with $X$-degree $d\geq 2$. Define

$$L_f(n):=\operatorname{lcm}\left(f(Q):Q\in\mathbb{F}_q[T]\text{ monic},\ \deg Q=n\right).$$

Let $V_f:=\{g\in\mathbb{F}_q[T]:f(X+g)=f(X)\}$ and $c_f:=1/|V_f|$. The function-field Cilleruelo conjecture. As $n\to\infty$,

$$\deg L_f(n)\sim c_f(d-1)nq^n.$$

This is the function-field analogue of Cilleruelo's conjecture. The source presents it as a conjecture; the supplied material gives no resolution status.