Ye–Lim's universal Toeplitz factorization conjecture

Let $n\in\mathbb{N}$ and let $M\in\mathbb{C}^{n\times n}$. Write $\mathrm{Toep}(M)$ for the minimal number of Toeplitz matrices whose product is $M$, and let $\mathrm{Toep}_n$ be the maximum of $\mathrm{Toep}(M)$ over all $n\times n$ matrices.

Ye–Lim's conjecture. For all $n\in\mathbb{N}$,

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

This asserts that the generic upper bound established by Ye and Lim holds for every matrix, not merely for generic matrices. The paper notes that the conjecture is true for $n=1,2$ but is disproved for $n=3$; the proposed replacement in odd dimensions appears later in the paper.