The unique gridding conjecture for coils in cyclic partial multiplication matrices

Let $M$ be a cyclic partial multiplication matrix containing $\ell$ non-empty cells. A coil is a permutation represented by a cyclic sequence of points following the cycle of non-empty cells of $M$. Unique gridding conjecture. Every coil of length at least $2\ell+1$ has only one $M$-gridding.

This conjecture proposes an improved bound over the previously established bound $(\ell+1)\ell^2+1$ for unique griddings of sufficiently long coils. The authors state that the bound $2\ell+1$ is suggested by consideration of short coils on relatively small cycles; no proof or resolution is supplied here.