The layered-permutation exponential-growth conjecture

For $w\in S_n$, let $\Upsilon_w=\mathfrak{S}_w(1^n)$, and let “$w$ layered” mean that $w$ is a layered permutation. Define

$$L_n=\max_{w\text{ layered}}\Upsilon_w,\qquad M_n=\max_{w\in S_n}\Upsilon_w.$$

Layered exponential-growth conjecture. The two maxima have the same exponential growth rate:

$$\lim_{n\to\infty}\frac{1}{n^2}\log_2 M_n = \lim_{n\to\infty}\frac{1}{n^2}\log_2 L_n \approx 0.29.$$

Equivalently, any improvement over layered permutations is subexponential in $n^2$.

The value of the layered limit was computed by Morales, Pak, and Panova, while the equality with the unrestricted limit is not proved in the source. The paper presents this as an expectation, so its status is open.