Asymptotic conjecture for products of cross-Sperner families

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{P}([n])$ be its power set, and let $\pi(n,k)$ denote the maximum of $\prod_{i=1}^k|\mathcal{F}_i|$ over all cross-Sperner $k$-tuples $(\mathcal{F}_1,\ldots,\mathcal{F}_k)$ in $\mathcal{P}([n])$. Here $o(1)$ tends to zero as $n\to\infty$ with $k$ fixed.

Asymptotic product conjecture. For fixed $k\geq 2$ and sufficiently large $n$ with respect to $k$,

$$\pi(n,k)=(1+o(1))\left(\frac{(k-1)^{k-1}}{k^k}2^n\right)^k.$$

This would determine the asymptotically sharp product of the sizes of a fixed number of cross-Sperner families. The source presents it as a belief in the closing remarks and does not state any resolution.