We've been made aware of an issue with the old 3.4.3.0/3.43 version (the last one that Marginal released) when checking for an update. It finds that there's a new version, but then crashes shortly after trying to offer it to the user (the changelog pane is blank and the entire window, along with the EDMarketConnector.exe process, disappears 2-3 seconds later).
For now we're advising all 3.4.3.0 users (who want to update, if you have any third-party plugins they'll likely need updating to work with 4.x versions) to follow the procedure outlined in You really do want to update to the 4.x series, else you're missing out on all sorts of bug fixes and enhancements.
As per I've contacted Marginal to see if he can change the redirect of the old update URL to point to a file I can directly control (rather than to the live GitHub one, where edits will affect all EDMC users). If he enacts that change then I'll experiment to see if changing the contents of the file will allow 3.4.3.0 to update.
