= Solution
In the X eigenbasis, $M$ is $\operatorname{diag}(I,X)$ on its physical indices. A physical X insertion changes the sign of the minus block and can be pulled through as
$$
X_{\rm phys}M=Z_{\rm virt}MZ_{\rm virt}=MX_{\rm phys}.
$$
Conjugating both physical legs by Z swaps the two blocks:
$$
Z_{\rm phys}MZ_{\rm phys}
=M X_{\rm virt}
=X_{\rm virt}M.
$$
Hence
$$
\boxed{S=T=X}.
$$
Likewise $N=\operatorname{diag}(I,Z)$. Its analogous relations are
$$
X_{\rm phys}N=X_{\rm virt}NX_{\rm virt}=NX_{\rm phys},
$$
$$
Z_{\rm phys}NZ_{\rm phys}
=NZ_{\rm virt}
=Z_{\rm virt}N.
$$
Solved by gpt-5.6-sol high.
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