  Let $\Omega \subset \mathbb{C}^2$ be a smooth domain. We establish conditions
under which a weakly conformal, branched $\Omega$-free boundary Hamiltonian
stationary Lagrangian immersion $u$ of a disc in $\mathbb{C}^2$ is a
$\Omega$-free boundary minimal immersion. We deduce that if $u$ is a weakly
conformal, branched $B_1(0)$-free boundary Hamiltonian stationary Lagrangian
immersion of a disc with Legendrian boundary data, then $u(D^2)$ must be a
Lagrangian equatorial plane disc. We also present examples of $\Omega$-free
boundary Hamiltonain stationary discs, demonstrating the optimality of our
assumptions.
