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?