Merzon–Smirnov's layered-permutation conjecture for maximal Schubert specialization

For $n\geq 1$, define

$$\alpha_n=\max\{\nu_w\colon w\in S_n\},$$

where $\nu_w=\mathfrak{S}_w(x)|_{x_i=1}$. A layered permutation $w(b_1,\ldots,b_k)$ is the concatenation of decreasing blocks on consecutive intervals, where $b_1+\cdots+b_k=n$. Merzon–Smirnov's conjecture. For $n\geq 1$, the permutations in $S_n$ attaining $\alpha_n$ are layered permutations. The conjecture identifies the maximizers of the principal specialization; the source presents it as a prediction of Merzon and Smirnov.