Sufficiency of the distinguished-point condition in defect cobordisms for the word problem

Ascertain whether the distinguished-point condition (which fixes a basepoint on defect circles) is sufficient to prevent uncontrolled wrapping of 1-defects under cobordisms in Bord_2^{def}(𝓟_G) when modeling the group word problem, or identify additional topological constraints on the underlying surfaces that are required.

Background

In translating the word problem into defect-cobordisms, the authors fix a basepoint on each defect circle to read words consistently. However, during bordisms, 1-defects might wrap around surfaces in ways that could undermine this control.

This open question asks whether the current basepoint rule (the distinguished-point condition) suffices, or whether further constraints are necessary to maintain a well-behaved theory of word-cobordisms.