Let $L$ be an asymptotically conical SL $m$-fold in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3$ with cone $C$ and rate $\rho\in(2-m,0),$ and write ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ for the moduli space of deformations of $L$ as an AC SL $m$-fold in ${\mathbin{\mathbb{C}}}^{m}$ with cone $C$ and rate $\rho$. Then ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ is a smooth manifold of dimension $b^{1}_{\rm cs}(L)=b^{m-1}(L)$.