Thank you @outlog your answer helped me debug the specific problem communicating
LiveView -> LiveComponent 
In parallel I also tried to solve a tangential problem here, which tries to have a common source of truth for all LVs that will consume the shoppingCart (basically I wanted to keep the localStorage-resumed shoppingCart in memory so every newly mounted LV could consume it without having to resume form localStorage again). Unfortunately I couldn’t land it in the way I intended, but I think I’m overcomplicating things so I’ll pivot my approach.
Regardless that’s a different story.
