Moore–Penrose invertibility transfer between zero-product and invertible triple products

Determine whether, for every *-ring R, the implication that abc=0 entails acbin R^{\dag} for all a,b,cin R implies that abcin R^{\dag} entails acbin R^{\dag} for all a,b,cin R.

Background

In Section 3, the paper considers four conditions on a *-ring R involving symmetry, projections, and Moore–Penrose invertibility of transposed triple products. Condition (3) asserts that abc=0 implies acb is Moore–Penrose invertible for all a,b,c in R, whereas Condition (4) asserts that Moore–Penrose invertibility of abc implies Moore–Penrose invertibility of acb for all a,b,c in R.

The paper establishes the implications that symmetry implies Condition (4), and that Condition (4) implies Condition (3). It also provides an example showing that the converses of the implications from symmetry to Conditions (3) and (4) do not hold in general. The unresolved issue is whether the weaker zero-product implication in Condition (3) is nevertheless sufficient to yield the stronger Moore–Penrose invertibility-transfer property in Condition (4).