A product version of the Hilton-Milner Theorem II

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\dots ,n\}$ are called {\it non-trivial cross-intersecting} if $F\cap G\mathrm{\ne }\mathrm{\varnothing }$ for all $F\in \mathcal{F},G\in \mathcal{G}$ and $\cap \{F\text{\hspace{0.17em}⁣}:F\in \mathcal{F}\}=\mathrm{\varnothing }=\cap \{G\text{\hspace{0.17em}⁣}:G\in \mathcal{G}\}$. In this note, we establish the product version of the Hilton-Milner Theorem for $k\ge 8$ in the full range. That is, if $\mathcal{F},\mathcal{G}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ are non-trivial cross-intersecting, $n\ge 2k+1$ and $k\ge 8$, then $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le {(\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0px}{}{n-k-1}{k-1}\right)+1)}^2\mathrm{.}$$