Which AC-3 algorithm is being used here?
The Fig. 5 in the question depicts AC-algorithm with backtracking.

In fact, without backtracking, checking arc-consistency alone can hardly solve the 4-Queen problem. Unless it happens that choices made so far will be extended to a desired configuration, any algorithm to solve CSP have to backtrack so as to check other possibilities.

Here is a relevant excerpt from the article.


  Conversely, if it does not become possible to place a Queen at  the  pointed  column,  like  in  Fig. 5(c),  backtracker  will backtrack  the  previous  assigned  Queen’s  column  to  reassign the conflicted Queen to a newly safe cell at the same column. Then, it will, at first, clear the constraints it created by calling the implemented modular function.