As of the latest testing with AnyConnect, these issues were noticed under Big Sur:
• If a network adapter is removed (accidentally or otherwise) while the VPN is connected, reconnecting the adapter may require a reboot of the computer to be able to successfully reconnect to the VPN (bug filed with Cisco and it’s on their list to resolve.)• Remote network connections to Big Sur systems with AnyConnect installed may become occasionally unreachable to SSH or Apple Remote Desktop/VNC connections (bug filed with Apple about this.)
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