It turned out those window screen splines lasted a dozen years until this happened:

Some rummaging in the Big Box o’ String produced the spool of 1000 pound test Kevlar cord most recently applied to the seat back on Mary’s bike, so this happened:

Having re-confirmed that frayed Kevlar cannot be melted into a blob, another UV-stabilized cable tie at each end will control those tufts.
Those cords should last forever …
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