Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and $L$ a compact Lagrangian $m$-fold in $M$ with isolated conical singularities modelled on stable SL cones in ${\mathbin{\mathbb{C}}}^{m}$ (with any phase $e^{i\phi}$). Then for small $\epsilon>0$ there exists a unique smooth family $\{L^{t}:t\in[0,\epsilon)\}$ satisfying Lagrangian MCF with $L^{0}=L,$ where the $L^{t}$ are compact Lagrangians in $M$ with stable isolated conical singularities.