More evidence is provided for the conjectured correspondence between the D3-brane action in AdS₅ × S⁵ and the low-energy effective action for = 4 SU(N) SYM on its Coulomb branch, where the gauge group SU(N) is spontaneously broken to SU(N−1) × U(1) and the dynamics is described by a single = 2 vector multiplet corresponding to the U(1) factor of the unbroken group. Using an off-shell formulation for = 4 SYM in = 2 harmonic superspace, within the background-field quantization scheme we compute the two-loop quantum correction to a holomorphic sector of the effective action, which is a supersymmetric completion of interactions of the form Ω((F⁺)²|Y|⁻⁴)(F⁺)²(F⁻)²|Y|⁻⁴, with F^± the (anti) self-dual components of the U(1) gauge field strength, and Y the complex scalar belonging to the vector multiplet. In the one-loop approximation, Ω was shown in [13] to be constant. It is demonstrated in the present paper that Ω(F⁺)²|Y|⁻⁴ at the two-loop order. The corresponding coefficient proves to agree with the F⁶ coefficient in the D3-brane action, after implementing the nonlinear field redefinition which was sketched in [4] and which relates the = 2 vector multiplet component fields with those living on the D3-brane. In the approximation considered, our results are consistent with the conjecture of [4] that the = 4 SYM effective action is self-dual under = 2 superfield Legendre transformation, and also with the stronger conjecture of [5] that it is self-dual under supersymmetric U(1) duality rotations.