This could cause a drama about questioning specific user "has logged in this forum, but refused to answer" or "hasn't logged in this forum".
Usually when there's a problem about specific user, people would like to wait till the user logged in and let him to defend himself before tagging him. Now, if someone tagged the user when he didn't receive the reply for few days or weeks, people might disagree and say you're tagged too fast.
It will also be a little harder to detect those cheating Alt accounts using the "Last active time" metric if they decided not to show others their online status.
Using last active time metric to detect alt accounts is really pointless, no one gonna believe it.
Most people will ask "solid evidence" if you fail to bring that even though you're sure they're alts, people will say it's just your own assumption/suspect without any proof.