When PlexusTerm starts it asks GitHub for the latest published release, once. If it is newer than the version you run, a bar under the toolbar says so: Download opens the release page in your browser, where you pick the package for your system, and the close button hides the bar until the next start. PlexusTerm never downloads or installs anything by itself, and a failed check (no network, GitHub unavailable) shows nothing.
Settings → About has Check for updates to ask at any time; it says whether PlexusTerm is up to date, offers Download for a newer release or shows why the check failed. Turn off Check for updates at startup there to stop the startup check; the only data sent is a request for the latest release of the public repository, with PlexusTerm's version as the user agent.