k-multi-layered maximizer conjecture at a root-of-unity specialization

Fix $k\in\mathbb{Z}_{>0}$, let $\zeta$ be a $k$-th root of unity, and let $\Phi_w^k$ denote the principal specialization $\mathfrak{S}_w(1,q,q^2,\ldots)$ at $q=\zeta$. Define

$$u_k(n):=\max_{w\in S_n}\Phi_w^k,$$

and let $v_k(n)$ be the corresponding maximum over $k$-multi-layered permutations. k-multi-layered maximizer conjecture. Given $k\in\mathbb{Z}_{>0}$, for every $n$, all permutations reaching the maximum $u_k(n)$ under principal specialization $\mathfrak{S}_w(1,q,q^2,\ldots)$ at $q=\zeta$ are $k$-multi-layered permutations. In other words,

$$u_k(n)=v_k(n).$$

Thus, there is a limit

$$\lim_{n\to\infty}\frac{\log_2\nu_k(n)}{n^2}=\frac{\gamma}{k\log 2}.$$

The paper proves matching asymptotics for the restricted maximum $v_k(n)$ and gives general upper and lower bounds for $u_k(n)$; the unrestricted maximization statement and its limiting formula remain open.