For Banach spaces $E$ and $F$ we denote the approximate operators by $\mathcal A(E,F)$ and projective tensor product by $\hat\otimes$.
Consider the natural map $$\Delta: \mathcal A(\ell^q,\ell^p)\hat\otimes\mathcal A(\ell^p,\ell^q)\rightarrow \mathcal A(\ell^p), \quad S\otimes T\mapsto TS$$$$\Delta: \mathcal A(\ell^q,\ell^p)\hat\otimes\mathcal A(\ell^p,\ell^q)\rightarrow \mathcal A(\ell^p), \quad S\otimes T\mapsto ST$$
Can we write elements in image of $\Delta$ in form of $TS$ where $T\in\mathcal A(\ell^q,\ell^p)$ and $S\in\mathcal A(\ell^p,\ell^q)$?