Free boundary Hamiltonian stationary Lagrangian discs in C²

Let Ω ⊂ C² be a smooth domain. We establish conditions under which a weakly conformal, branched Ω-free boundary Hamiltonian stationary Lagrangian immersion u of a disc in C² is a Ω-free boundary minimal immersion. We deduce that if u is a weakly conformal, branched B₁(0)-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary data, then u(D²) must be a Lagrangian equatorial plane disc. We also present examples of Ω-free boundary Hamiltonain stationary discs, demonstrating the optimality of our assumptions.