Asymptotic sector-length bound for quantum states

Let $A_k$ denote the $k$-sector length of an $n$-qubit quantum state. Asymptotic sector-length conjecture. For all $k$ there exists an $n_0$ such that, for all $n\geq n_0$, every $n$-qubit state satisfies

$$A_k\leq\binom{n}{k}.$$

The bound is known for the cases established earlier in the paper, while numerical evidence suggests a similar statement for $A_4$ when the number of qubits is at least $8$. The authors note that shadow inequalities alone are insufficient to prove the conjecture in that case, and the general assertion remains open.