2-positivity to 3-positivity

Question

Let $B\in M_3(\mathbb{C})$ and $S_3= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} $. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$-by-$2$ compressions $X$ of $S_3$. Can we conclude $I_3\otimes I_3+B\otimes S_3+B^*\otimes {S_3}^*\geq 0$?

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