Hi kiwitek,
I took a quick look at the page and there are no Advanced Sidebar Menu being called on it.
Based on what you are describing, it sounds like there is something going on with either a server or site cache. As this plugin does not do any form of page/widget versioning it would have no way to revert your site back to old widgets and can be ruled out as the issue.
If your admin is showing the widget registered, I would look into any page cache or varnish type setup you have running. If the widgets are also disappearing from the the admin, I would look at any form of database or object cache.
Another possibility is a DNS issue, in case you have more than one environment running, it’s possible your browser is intermittently pointing to different environments. If you only have one environment, you can ignore this possibility. If you have more than one doing a quick ping may tell you more about what environment you are pointed to.
Obviously, as you have seen when using Incognito, be sure to flush your browser cache during testing to rule that out.
Hope this helps!
Mat Lipe