The AnyConnect VPN software offered by the VPN servers for Automatic upgrade for Mac users is now at 4.9.04043, which is the latest version provided by the vendor, and has their best Big Sur (Mac OS 11) support. There continue to be discussions at among people that run VPNs for Mac users that even this version does not work 100% reliably with Big Sur. So the VPN team continues to suggest that if VPN is critical for someone's work, to not upgrade to Big Sur until there is better support in the Cisco AnyConnect VPN client.