Leave $ T in M_n $. Is the following true?

$$ bigcap limits_ {B in M_2 \ text {tr} (B) = 0} left {X in M_2: W (X) subseteq W (T) text {y} W ( B otimes X) subseteq W (B otimes T) right } subseteq bigcap limits_ {B in M_2} left {X in M_2: W (B otimes X) subseteq W (B otimes T) right }? $$ where $ W (S): = { langle Sx, x rangle: Vert x Vert = 1 } $ it's called numerical range of $ S $.

**Comments:** I have checked the above in MATLAB using some particular option of $ T, B $ which is affirmative for those cases. So, I have tried to prove the above as follows:

- We know $ W (X) subseteq W (T) $ if a map $ varphi: text {span} {I, T, T ^ * } rightarrow text {span} {I, X, X ^ * } $ S t. $ varphi (aI + bT + cT ^ *) = aI + bX + cX ^ * $ where $ a, b, c in mathbb {C} $ is positive. So, by hypothesis, we have a positive map that says $ psi: text {span} {I_2 otimes I, text {span} {B _ { circ}, B _ { circ} ^ * } otimes text {span} {T , T ^ * } } rightarrow M_4 $ S t.$ psi (I_2 otimes I + B otimes T + B ^ * otimes T ^ *) = I_2 otimes I_2 + B otimes X + B ^ * otimes X ^ * $ it's positive where $ B in M_2 $ S t $ tr (B) = 0 $ Y $ B _ circ}

begin {pmatrix}

0 and 1 \

0 and 0 \

end {pmatrix}.

$ Now, I have no idea if the map $ varphi $ can extend positively to $ text {span} {I, B _ { circ}, B _ { circ} ^ * } otimes text {span} {I, T, T ^ * } $ to get the required result.

It may be false, but I still don't have any counterexample. Thanks in advance. Any comments are greatly appreciated.