This is the result of removing the last segment of wire and redefining the Fixed Joint to holding the next vertex. The other change was to set the Initial and Minimum Substeps to 100 and the Maximum Substeps to 1000.
Notice that the solver can’t get started, and has bisection before the first converged substep after 5 iterations with this warning:
EQUIL ITER 5 COMPLETED. NEW TRIANG MATRIX. MAX DOF INC= -0.2124
*** WARNING *** CP = 7.641 TIME= 21:22:20
Contact element 85693 (real ID 10) status changes abruptly from contact
(with target element 86405) -> no-contact.
Below is the displacement of the first converged substep. Notice that the wire has left the hole at the very first substep
The Displacement condition shown below leaves the end of the wire Free in two directions. This is a bad idea. The wire is being pulled in a straight line along that local Y axis. Why not fix those two directions at 0, that would be a more stable configuration.
Making this suggested change results in the wire staying in the hole.