Update (5/29/2018): Over the past several days, some users have reported Could not authenticate and other errors in Tweetlogix. The problem has turned out to be an issue with the app related to the way it saves and restores tweets. This issue affects users of iOS 10 and earlier and is unrelated to the recent Tweetlogix 2.14 update. We're planning to submit an update soon to resolve the issue for iOS 10 users.
For now, a possible workaround is to upgrade your device to iOS 11, however you will need to clear your Tweetlogix Cache (in Tweetlogix Settings) after upgrading.
Sorry about this problem. Thanks to everyone who reported the issue and provided details that helped diagnose it.