Is there method to manage the visited state of the default menu?

1)  Articulate doesn't provide a way to adjust when the tick mark appears on the built-in Menu. A slide is marked when its timeline starts. (That behavior could probably be altered with JavaScript, but I can't help with that.)

2)  You can track completion of any slide using a variable. For example, create a T/F variable with the default value of False. Add a trigger that changes the variable to True when the slide/interaction is complete.

• The "when" in the trigger might be when the timeline ends, when designated objects are in a Visited state, when the user reaches the final layer in a Carousel activity, or whatever means the slide is completed. 
• A variable's value stays the same until a trigger changes it. And the value can be accessed from any slide. That means you could use the value of a slide's T/F variable to control what happens elsewhere in the course.

FYI: The interactive slides in your Sample file have a trigger to disable the Next button when the timeline starts.

• That trigger will run every time the user visits the slide, which means the Next button will be disabled even if they already completed the interaction on a previous visit. 
• To prevent that from happening, the trigger needs some kind of condition(s). 
• This post has a demo for different ways to control the Next button: TIP: Controlling the NEXT Button 101 - Articulate Storyline Discussions - E-Learning Heroes

It’s worth the effort to learn about variables and trigger conditions, because they provide the real power in Storyline. Here’s more info:

• https://community.articulate.com/discussions/articulate-storyline/the-value-of-variables
• https://community.articulate.com/series/articulate-storyline-360/articles/articulate-storyline-360-user-guide-how-to-work-with-triggers#conditions 

Here i show a solution for this.
What you would need to get it working in your setup is some conditions ( either plain Storyline or Javascript ) that decide when the menuitems need to be restricted.

https://360.articulate.com/review/content/c6b7e48a-f204-441d-81ea-4cdd5ed6a524/review

One big plus of this approach is that you can customize the menu/listitems if you want to.
https://360.articulate.com/review/content/146cbdd5-f36c-4ce4-a5f6-94effde85c42/review

As you can see in the second review.
And here is the working Storyline.

Kind regards,
Math

Could you please provide guidance on how to select different elements? I've noticed that the same selector has been used for all list items.

Please refer to the attachment. 

As is this would been easy to show...but as Articulate changes HTML-structure without notice on whats changing on updates, the recent update did this again, and now this doesnot work anymore :-(

We are working on a tutorial-app/website to show and help with things like this. Will be sharing it when live. And will workout a solution for this as one of the tutorials.

Correction: original still works... mistake in my code ;-) Will get a working sample of this in the tutorial-app...