Extend the restricted-sumset lower bound under a linear size condition

Prove that for every constant c>0, finite subsets A,B of the finite field F_p with |B|≤|A| and |A|+c|B|<p, and an arbitrary function f:B→A defining the forbidden matching relation R={(f(b),b):b∈B}, satisfy |A⧹+_R B|≥|A|+|B|−3.

Background

The paper confirms Lev’s conjecture for restricted sumsets arising from matchings when |A|+|B|≤(1−ε)p and p is sufficiently large, subject to an additional lower bound |B|≥c_εp in one formulation. The authors seek a stronger result in which the condition controlling the relative sizes of A and B is linear, such as |A|+c|B|<p.

The stated obstacle is that the parameter c_ε obtained in the main finite-field theorem is not linear in ε. Establishing the bound under an assumption of the form |A|+c|B|<p would imply the subsequent conjectured extension for arbitrary functions from B to A.