Let $T\in\mathscr{B(\mathcal{H})}$ and $A,B\in M_n(\mathbb{C})$ where $A$ is a hermitian matrix s.t. $$\lambda_1(A\otimes I_n+B\otimes X+B^*\otimes X^*)\leq \lambda_1(A\otimes I+B\otimes T+B^*\otimes T^*).$$ Does there exist a hermitian $C\in M_n(\mathbb{C})$ s.t. $$\lambda_1((A+C)\otimes I_n+B\otimes X+B^*\otimes X^*)\leq \lambda_1((A+C)\otimes I+B\otimes T+B^*\otimes T^*)$$ with $(A+C)B=B(A+C)$?
Notations: Let $S\in\mathscr{B(\mathcal{H})}$ be a selfadjoint operator. Then $\lambda_1(S):=\sup\limits_{\Vert x\Vert=1}\langle Sx,x\rangle$. Note that whenever $S$ is a hermitian matrix, $\lambda_1(S)$ is the largest eigen value of $S$.