On Freiman's Theorem in a function field setting

We prove some new instances of a conjecture of Bachoc, Couvreur and Z\'emor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if F is a rational function field over an algebraically closed field K and S ⊂ F a finite dimensional K-vector space such that S² = 2 S + 1, then the conjecture holds.