The linear-factor conjecture for Wie-length

Let $S$ be a finite generating system of $M_n(\mathbb{C})$, with $\ell(S)$ its ordinary length and $\operatorname{Wie}\ell(S)$ its Wie-length. Linear-factor conjecture for Wie-length. One should have

$$\ell(S)\leq \operatorname{Wie}\ell(S)=O\bigl(n\,\ell(S)\bigr).$$

The lower bound is immediate, while the conjectured improvement concerns the factor relating the two lengths. The paper motivates this by examples with $\ell(S)=\Theta(n)$ and $\operatorname{Wie}\ell(S)=\Theta(n^2)$, and notes that the currently established general comparison has an extra factor of order $n$.