Free Boundary Hamiltonian Stationary Lagrangian Discs in

Let $$ be a smooth domain. We establish conditions under which a weakly conformal, branched $$ -free boundary Hamiltonian stationary Lagrangian immersion u of a disc in $$ is a $$ -free boundary minimal immersion. We deduce that if $$ is a weakly conformal, branched $$ -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then $$ is a Lagrangian equatorial plane disc. Furthermore, we present examples of $$ -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.