Compare centered and uncentered few-moment realignment criteria

Determine whether, for every nonzero positive semidefinite bipartite operator with nonzero centered-normalization factor, the condition that the normalized centered realignment moments satisfy \(\alpha(r(\tilde X/\sqrt{G(X)}))\leq1\) implies that the normalized uncentered realignment moments satisfy \(\alpha(r(X/t_X))\leq1\).

Background

The enhanced realignment criterion is shown earlier to imply the standard realignment criterion at the level of trace norms. The paper then introduces the discrete moment functional $\alpha$, which gives optimal trace-norm information from finitely many even singular-value moments.

Whether the same implication persists after replacing the full trace norms by the corresponding finite-moment conditions is left unresolved. The question concerns arbitrary positive semidefinite bipartite operators, subject to the nonzero assumptions needed to define the two normalizations.