A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces

We construct a linear operator $T:\mathscr S'(\mathbb R^n)\to \mathscr S'(\mathbb R^n)$ such that $T:\mathscr B_{pq}^s(\mathbb R^n)\to\mathscr B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but $T(\mathscr F_{pq}^s(\mathbb R^n))\not\subset \mathscr F_{pq}^s(\mathbb R^n)$ unless $p=q$. As a result Triebel-Lizorkin spaces cannot be interpolated from Besov spaces unless $p=q$. In the appendix we purpose a question for the interpolation framework via structured Banach spaces.