Suppose $L$ is a closed, embedded, exact, asymptotically conical special Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3,$ asymptotic at rate $\rho<0$ to a union $\Pi_{1}\cup\Pi_{2}$ of two transversely intersecting special Lagrangian planes $\Pi_{1},\Pi_{2}$ in ${\mathbin{\mathbb{C}}}^{m}$. Then $L$ is equivalent under an $\mathop{\rm SU}(m)$ rotation to one of the ‘Lawlor necks’ $L_{\boldsymbol{\phi},A}$ found by Lawlor [45], and described in Example 2.5.