The quadratic injectivity-threshold conjecture

Let $S=\{A_1,\ldots,A_g\}\subseteq M_n(\mathbb{C})$ satisfy condition $C1$, meaning that words of some common length span $M_n(\mathbb{C})$, and let $\Gamma_L$ be the associated map. Quadratic injectivity-threshold conjecture. There exists a function $L(n)=\Theta(n^2)$ such that $\Gamma_L$ is injective for every $L\geq L(n)$ and every such set $S$. This is the matrix-product-state formulation of the worst-case quantum Wielandt conjecture. The paper notes that the best known general bound has an additional logarithmic factor, so this formulation remains open.