(Sketch) Assume the suppports do not coincide. After small Hamiltonian perturbations, we can ensure the perturbed Lagrangians $\tilde{L},\tilde{L}^{\prime}$ have transverse intersections, and still define the same isomorphic objects in $D^{b}Fuk(X)$. Under the special Lagrangian assumption, through judicious choice of the Hamiltonian via Morse theory as in Thomas-Yau [66, Thm 4.3], or by using genericity arguments based on real analyticity as in Joyce-Imagi-Santos [40, section 4.3], one can ensure there is no intersection point $\tilde{L}\cap\tilde{L}^{\prime}$ of degree $0,n$, so in particular $HF^{0}(L,L^{\prime})\simeq HF^{0}(\tilde{L},\tilde{L}^{\prime})=0$. However, this implies the cohomological unit of $HF^{*}(L,L)$ is zero, so the Floer cohomology ring of $L$ is zero, namely $L$ is a zero object in $D^{b}Fuk(X)$, contradiction. ∎