I use a black diamond Alpine Cork hiking pole for all my trips (one pole only---keeps my pruner arm and hand free)---and just got back from a trip whereby the cork on the pole chunked off . . . once again. I'm on my third such BD pole. (Made a mistake 10 years ago buying 2 sets of poles).
Your gossamer fake corks don't look they are gonna fit as the top handle knob might have to be hacksawed off and the cork hole reamed out.
I've done two things to replace this cork---get foam walking cane grips and either cut one lengthwise and wrap around the pole where the cork used to be---using super glue and a long string to keep it tight while drying---
Or cut off the handle knob as mentioned and slide a foam grip down on the handle shaft.