Sárközy'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 $A\subseteq \{f\in {\mathbb{F}}_q[t]\text{\hspace{0.17em}⁣}:\text{deg }f\le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green’s approach to Sárközy’s theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that $$\mathrm{\mid }A\mathrm{\mid }\ll q^{(N+1)(11\mathrm{/}12+o(1))},$$ improving upon the bound $O\left(q^{(1-c\mathrm{/}\log N)(N+1)}\right)$ due to Lê and Spencer.