Bounded invertible reduction conjecture for Toeplitz factorizations

Let $\mathrm{Toep}_n^{\mathrm{Inv}}$ denote the minimal number of Toeplitz factors needed to factor every invertible $n\times n$ matrix. A matrix is understood to be factored as a product of matrices, and $\mathrm{Toep}_n$ is the maximal Toeplitz number among all $n\times n$ matrices.

Bounded invertible reduction conjecture. There exists a constant $C>0$ such that every matrix can be expressed as a product of an invertible matrix and $C$ Toeplitz matrices. In particular,

$$\mathrm{Toep}_n\leq\mathrm{Toep}_n^{\mathrm{Inv}}+C.$$

This would reduce the general factorization problem to the invertible case up to an additive constant. The paper proves that $C=3$ suffices for $n=3$ and $n=4$, while the existence of a dimension-independent constant remains open.