Is this braided cord authentic for AE 40?

[countryman]

Good detail and nice clear photos. Thanks for adding.

Yes, indeed. Those details are good to know when starting restoring phones...
The company made elasticords to fit various phone models, supposedly they used the same basic design but made the ends in different ways.
On a British braided (non-retractable) cord I found 3 conductors are used and one thread looks the same, but has no copper inside. It serves as a tie to fasten the ends, as a strain relief.
For a retractable cord it might be better to use 4 absolutely identical threads, so that it collapses in a symmetrical way?
Possibly on the shown phone an elasticord for a different model was used. It seems the phone requires a crimped strain relief, while the cord has tie ends. It may be accepted as historic though, and there are ways to adapt the strain relief so that the cord sits firmly as it should.