Allow one's mind to drift perhaps and falter as an embroiderer? (4,3,6)

I believe the answer is:

lose the thread

'allow one's mind to drift perhaps' is the definition.
(to lose the thread is to stop concentrating)

I cannot understand how the rest of the clue works.

(Other definitions for lose the thread that I've seen before include "A careless seamstress may" , "forget her train of thought?" .)

I've seen this clue in The Times.

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