Converse implication for squared operators

Determine whether injective non-negative self-adjoint operators S and T satisfying S^2\overset{\mathrm{w}{\sim}}T^2 and S∼T necessarily satisfy S^2∼T^2.

Background

The paper proves that S2∼T2 implies S∼T by applying its power-stability theorem. It then asks whether the reverse implication holds when S2 and T2 are assumed to be weakly equivalent.

Two special cases are established: the implication holds when S and T strongly commute and when T is bounded with a bounded inverse. The general case remains unresolved.

References

The converse may also be true: Let $S,T\in\mathcal{S!A}_+(\mathscr{H})$ satisfy $S2\overset{\mathrm{w}{\sim}} T2$ and $S\sim T.$ Is it true that $S2\sim T2$?

— Representation Theory of Canonical Commutation Relations Arising from the Quantization of the Klein-Gordon Equation with an External Potential  (2609.05237 - Matsuzawa et al., 4 Sep 2026) in Question in subsection “A related question,” Section 2, following Corollary \ref{gener of MSU}