CMS items inconsistent between Designer and Published site

Hello everyone,
I keep having issues with what I see in the Designer and Preview space of my website and what gets published when editing CMS items.
In the home page I have 3 items containing projects all attached to the same CMS collection (one collection per item, don’t know how to optimise it for now).
In the page Progect I have 5 elements, same issues.

So the collection is the same, but each item starts from a different element (Item 1: Show 1 - Start at 1; Item 2: Show 1 - Start at 2; and so on). The problem is, in the Designer and Preview I see the correct elements from the collection in the right position (Cover images, texts, paragraphs…). But when I publish it, elements starts mixing (Cover images of Item 1 together with Paragraph of item 2…).

I tried to delete cookies from browsers and it only worked once. I changed browser after publishing but the issues remains. Any suggestion on how to solve this?
See screenshots.
Thank you.

Here is my public share link: LINK
(how to access public share link)

First screen is from the Designer, second is the published site. You see the second item goes in third position on the far right, the project Phygital Library is not even there and the central position is taken by another project (that is at position 4 in the CMS collection)