Super-exponential growth of symmetric chain decompositions of $L(m,n)$

Let $L(m,n)$ be the minuscule lattice of partitions contained in an $m$ by $n$ box, and let $\#\operatorname{SCD}(L(m,n))$ denote the number of symmetric chain decompositions of this poset. Fix $m>1$ and $k\in\mathbb{N}$. Super-exponential growth conjecture for $L(m,n)$.

$$\lim_{n\rightarrow\infty}\frac{\#\operatorname{SCD}(L(m,n))}{k^n}=\infty.$$

This asserts that, for each fixed $m>1$, the number of symmetric chain decompositions of $L(m,n)$ eventually exceeds every exponential function $k^n$. The conjecture is based on computational data; the existence and enumeration of symmetric chain decompositions remain active problems.