Topological interpretation of the weak braiding operad wPaB

Characterize the relationship between the weak braiding operad wPaB (weakly unital parenthesized braids) and a suitable configuration-space model, specifically a configuration space of universal covers of punctured discs, paralleling the established connection between the operad PaB and configuration spaces of discs.

Background

The operad of parenthesized braids PaB is classically tied to configuration spaces of discs and encodes braided monoidal structures. The paper introduces the operad wPaB to capture weak-braided structures.

This question seeks a topological or geometric model for wPaB analogous to that of PaB, potentially in terms of configuration spaces of universal covers of punctured discs, which could elucidate the homotopical or geometric origins of weak braiding.