Suppose $L$ is a closed, embedded, exact, asymptotically conical Lagrangian MCF expander in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 2,$ satisfying the expander equation $H=\alpha F^{\perp}$ for $\alpha>0,$ and asymptotic at rate $\rho<2$ to a union $\Pi_{1}\cup\Pi_{2}$ of two transversely intersecting Lagrangian planes $\Pi_{1},\Pi_{2}$ in ${\mathbin{\mathbb{C}}}^{m}$. Then $L$ is equivalent under a ${\rm U}(m)$ rotation to one of the LMCF expanders $L_{\boldsymbol{\phi}}^{\alpha}$ found by Joyce, Lee and Tsui [43, Th.s C & D], and described in Example 2.13.