Asymptotic conjecture for the maximal number of stamp foldings

For a $1\times n$ strip of stamps, let $M(n)$ be the maximal number of ways to fold the strip over all mountain-valley assignments. Maximal-folding asymptotic conjecture. As $n\to\infty$,

$$M(n)\sim \frac{2^n}{n^{5/4}}.$$

The source says this behavior is suggested by computational plots and is not proved there.