S\'ark\"ozy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\text{deg}~ f \le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to S\'ark\"ozy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that $$|A| \ll q^{(N+1)(11/12+o(1))},$$ improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to L\^{e} and Spencer.