Perez-Garcia–Verstraete–Wolf–Cirac stabilization conjecture

Let $L$ be a linear space of $D\times D$ matrices, and let $L^k$ denote the linear space spanned by all products of $k$ matrices in $L$. If

$$\dim L^k=D^2$$

for some $k$, then

$$\dim L^k=D^2$$

for all $k\geq cD^2$, for some constant $c$ independent of $D$. Perez-Garcia–Verstraete–Wolf–Cirac conjecture. The sequence $L^1,L^2,\ldots$ should stabilize after $O(D^2)$ terms. This conjecture is motivated by uniform Matrix Product States and is a quantum analogue of Wielandt's inequality. The paper improves the previously known $O(D^4)$ bound to $O(D^2\log D)$, but the stated $O(D^2)$ conjecture is not identified as resolved here.