I seem to recall from testing, that caching of sudo is independent of how you authenticate -- but it's been so long since I played with it, that I'm not certain.
"sudoers uses per-user time stamp files for credential caching. Once a user has been authenticated, a record is written containing the uid that was used to authenticate, the terminal session ID, and a time stamp (using a monotonic clock if one is available). The user may then use sudo without a password for a short period of time (5 minutes unless overridden by the timeout option). By default, sudoers uses a separate record for each tty, which means that a user's login sessions are authenticated separately. The tty_tickets option can be disabled to force the use of a single time stamp for all of a user's sessions."
Note that the text uses "password", but I think it means "successfully authenticate". ]
[ed: Looks like it is independent: https://www.sudo.ws/man/sudoers.man.html
"sudoers uses per-user time stamp files for credential caching. Once a user has been authenticated, a record is written containing the uid that was used to authenticate, the terminal session ID, and a time stamp (using a monotonic clock if one is available). The user may then use sudo without a password for a short period of time (5 minutes unless overridden by the timeout option). By default, sudoers uses a separate record for each tty, which means that a user's login sessions are authenticated separately. The tty_tickets option can be disabled to force the use of a single time stamp for all of a user's sessions."
Note that the text uses "password", but I think it means "successfully authenticate". ]