Multilinear rough singular integral operators may not be of the weak type at the endpoint

We show that $m$-linear homogeneous rough singular integral operators ${\text{LL}}_{\mathrm{\Omega }}$ associated with $L^{\mathrm{\infty }}$ functions $\mathrm{\Omega }$ on ${\mathbb{S}}^{mn-1}$ with mean value zero may not be bounded from $L^1({\mathbb{R}}^n)\times \cdots \times L^1({\mathbb{R}}^n)$ to $L^{\frac{1}{m},\text{nf}}({\mathbb{R}}^n)$ except possibly for the case $n=1$ and $m=2$.