Commutation with non-zero compact operators for typical positive contractions on ℓ2

Determine whether a typical positive contraction in (P1(ℓ2), SOT) (and analogously in (P1(ℓ2), SOT*)) commutes with a non-zero compact operator on ℓ2.

Background

For general (not necessarily positive) contractions on ℓ2, it is known that a typical contraction does not commute with any non-zero compact operator. That argument uses unitary equivalence and does not extend to positive contractions.

The paper poses the positive-operator analogue for typical positive contractions under SOT (and SOT*).

References

Thus, the following question is open. Does a typical $T \in (, SOT)$ commute with a non-zero compact operator if $X = \ell_2$? And what about a typical $T \in (, SOT)$?

— Typical properties of positive contractions and the invariant subspace problem (2409.14481 - Gillet, 22 Sep 2024) in Section 5 (Further remarks and questions)