The number of symmetric chain decompositions

We prove that the number of symmetric chain decompositions of the Boolean lattice $2^{[n]}$ is $${(\frac{n}{2e}+o(n))}^{2^n}\mathrm{.}$$ Furthermore, the number of symmetric chain decompositions of the hypergrid $[t{]}^n$ is $$n^{(1-o_n(1))\cdot t^n}\mathrm{.}$$