What version of the plugin are you using? The screen shot does not have "verbose" option so there would likely not be anything in the jenkins.log file. If possible, can you upgrade to the latest version of the plugin, check the verbose option and then look at the jenkins.log to see if you get more details?
The test connection method is calling the Tower API endpoint: /api/v1/ping/ . The message "The item does not exist" happens with the plugin gets a 404 back from Tower which indicates a file not found message. Tower 3.2.2 does add a v2 of the api but v1 should not be deprecated. Can you call the /api/v1/ping/ endpoint with the Jenkins user that you provisioned in Tower?
Finally, the Tower URL you are specifying in the URL field, is that a direct connection to the tower server or is it going through some sort of proxy?
reiimer you are correct about the API transforming the urls to end with a /. In Jenkins, when you add a / to the end of the URL Jenkins ends up making calls to Tower like https://<server>//api/v2/ping (note the double slash at the start of the path in the URL. This is why the test fails with the message "The item does not exist". I just released an 0.8.3 version of the plugin which will string the ending / of the URL so the plugin should now work with a url of https://towerserver.something.com/ or https://towerserver.something.com. It normally takes a couple hours for the new version of the plugin to propagate. If you get a chance can you test this change and let me know if its now working either way for you?
alex_s can you test the 0.8.3 version of the plugin and let me know if this change now remediates the issue you were having?