Pick a connected graded Lagrangian $L$ whose obstructions [FO3] to the existence of its Floer cohomology vanish, and whose second Stieffel-Whitney class $w_{2}$ is the restriction of a class $\in H^{2}(X;\mathbb{Z}/2)$ on the whole manifold (for instance if $L$ is spin).

Then there can be at most one smooth special Lagrangian in the hamiltonian deformation class of $L$.

In particular, SLag homology spheres are unique in their hamiltonian deformation class in dimension 3 and above.