If the language of a TM is TMs which cannot self recognize, can the original TM?
You have to argue formally. Intuition can be helpful in pointing the way, but you need a formal argument in the end.

Hint: Suppose that $\langle N \rangle \notin L_s$. Then $N$ accepts $\langle N \rangle$, and so $\langle N \rangle \in L(N) \subseteq L_s$, a contradiction.