I have been trying to weight paint an umbrella to rig bones. However, I haven't been able to resolve the edges furthest away from the bones. They overlap when the umbrella is near closing. I have looked at this post for rigging an umbrella including the weight paint and I have tried to replicate it but I have not been successful. Would be really happy to hear suggestions of how to properly rig an umbrella and/or assign vertex weight paint colors.
link to my rig file: rig file link
Here are screenshots of my current weight painting vertex weights as they are in the screenshots above.
and here are working vertex colors from the link. Although, the umbrella I am working is upside down in normal open position, I don't understand why it doesn't work