Asymptotic conjecture for equal-block mountain-valley assignments

Let $S(m,k)$ denote the MV assignment with $m$ blocks, each of size $k$, and let $c(S(m,k))$ be its number of foldings. Equal-block asymptotic conjecture. For any fixed $m$ and sufficiently large $k$,

$$c(S(m,k))\sim \frac{e^m}{\sqrt{2\pi m}} k^m.$$

The conjecture gives a precise polynomial asymptotic for the equal-block family, based on computational experimentation; no proof or resolution is supplied in the source.