Ceiling-bound conjecture for the maximal Toeplitz number

Let $n>1$, and let $\mathrm{Toep}_n$ be the maximum, over all $n\times n$ matrices, of the minimal number of Toeplitz matrices required to factor the matrix.

Ceiling-bound conjecture. For $n>1$,

$$\mathrm{Toep}_n=\left\lceil\frac{n}{2}\right\rceil+1.$$

The paper proposes this as a replacement for Ye and Lim's conjecture after disproving the floor-bound conjecture in dimension $n=3$. It is presented as a reasonable substitution, and no resolution is supplied.